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UNIVERSITY   OF   CALIFORNIA 


Received      rU^HJ^t^lH^^^  ^  J  ^  f  9 
Accessions  No./..Q./..Q...        Book  No.  ../.*>. 


DEPARTMENT   OF   COMMERCE 


OF  THE 


BUREAU  OF  STANDARDS 

S.  W.  STRATTON,  DIRECTOR 

No.  74 

RADIO  INSTRUMENTS  AND 
MEASUREMENTS 


ISSUED  MARCH  23,  1918 


PRICE,  60  CENTS 

Sold  only  by  the  Superintendent  of  Documents,  Government  Printing  Office 
mgton,  D.  C. 


WASHINGTON 
GOVERNMENT  PRINTING  OFFICE 

1918 


\ 


TK57V/ 

CONTENTS 


PART  I.  THEORETICAL  BASIS  OF  RADIO  MEASUREMENTS 5 

Introduction 5 

The  fundamentals  of  electromagnetism 6 

1.  Electric  current i 6 

2.  Energy 9 

3.  Resistance n 

4.  Capacity 13 

5.  Inductance 14 

The  principles  of  alternating  currents 19 

6.  Induced  electromotive  force 19 

7.  Sine  wave 21 

8.  Circuit  having  resistance  and  inductance 23 

9.  Circuit  having  resistance,  inductance  and  capacity 25 

10.  "  Vector"  diagrams 28 

11.  Resonance 31 

12.  Parallel  resonance 39 

Radio  circuits 41 

13.  Simple  circuits 41 

14.  Coupled  circuits 45 

15.  Kinds  of  coupling 48 

16.  Direct  coupling : 52 

17.  Inductive  coupling 56 

18.  Capacitive  coupling 60 

19.  Capacity  of  inductance  coils. 62 

20.  The  simple  antenna 69 

21.  Antenna  with  uniformly  distributed  capacity  and  inductance.  ...  71 

22.  Loaded  antenna 75 

23.  Antenna  constants 81 

Damping 86 

24.  Free  oscillations 86 

25.  Logarithmic  decrement 90 

26.  Principles  of  decrement  measurement 92 

PART  II.  INSTRUMENTS  AND  METHODS  OP  RADIO  MEASUREMENT 96 

27.  General  principles 96 

Wave  meters 97 

28.  The  fundamental  radio  instrument 97 

29.  Calibration  of  a  standard  wave  meter 99 

30.  Standardization  of  a  commercial  wave  meter 104 

Condensers .' 108 

31.  General 108 

32.  Air  condensers no 

33.  Power  condensers 120 

34.  Power  factor 122 

35.  Measurement  of  capacity 129 

Coils 131 

36.  Characteristics  of  radio  coils 131 

37.  Capacity  of  coils 132 

38.  Measurement  of  inductance  and  capacity  of  coils 136 

Current  measurement 139 

39.  Principles 139 

3 


I    /  .      . 


4  Contents 

PART  II.  INSTRUMENTS  AND  METHODS  OF  RADIO  MEASUREMENT — Continued.  ' 

Current  measurement — Continued.  page 

40.  Ammeters  for  small  and  moderate  currents 141 

41.  Thermal  ammeters  for  large  currents 144 

42 .  Current  transformers 1 50 

43.  Measurement  of  very  small  currents 15  J' 

44.  Standardization  of  ammeters 170 

Resistance  measurement 175 

45.  High-frequency  resistance  standards 175 

46.  Methods  of  measurement 177 

47.  Calorimeter  method 177 

48.  Substitution  method 178 

49.  Resistance  variation  method 180 

50.  Reactance  variation  method 185 

51.  Resistance  of  a  wave  meter 187 

52.  Resistance  of  a  condenser 190 

53.  Resistance  of  a  coil 193 

54.  Decrement  of  a  wave 195 

55.  The  decremeter 196 

Sources  of  high-frequency  current 200 

56.  Electron  tubes 200 

57.  Electron  tube  as  detector  and  amplifier 204 

58.  Electron  tube  as  generator 2 10 

59.  Poulsen  arc 221 

60.  High-frequency  alternators  and  frequency  transformers 223 

61.  Buzzers 227 

62.  The  spark 228 

PART  III.  FORMULAS  AND  DATA 235 

Calculation  of  capacity 235 

63.  Capacity  of  condensers 235 

64.  Capacity  of  wires  and  antennas 237 

65.  Tables  for  capacity  calculations 241 

Calculation  of  inductance 242 

66.  General 242 

67.  Self-inductance  of  wires  and  antennas 243 

68.  Self-inductance  of  coils 250 

69.  Mutual  inductance 269 

70.  Tables  for  inductance  calculations 282 

Design  of  inductance  coils 286 

71.  Design  of  single-layer  coils 286 

72.  Design  of  multiple-layer  coils 292 

73.  Design  of  flat  spirals 296 

High-frequency  resistance 299 

74.  Resistance  of  simple  conductors 299 

75.  Resistance  of  coils 304 

76.  Stranded  wire 306 

77.  Tables  for  resistance  calculations 309 

Miscellaneous  formulas  and  data 312 

78.  Wave  length  and  frequency  of  resonance 312 

79.  Miscellaneous  radio  formulas 313 

80.  Properties  of  metals 317 

APPENDIXES 319 

Appendix  i. — Radio  work  of  the  Bureau  of  Standards 319 

Appendix  2. — Bibliography 324 

Appendix  3. — Symbols  used  in  this  circular 330 


RADIO  INSTRUMENTS  AND 
MEASUREMENTS 


PART   L— THEORETICAL  BASIS   OF  RADIO   MEAS- 
UREMENTS 


INTRODUCTION 

In  the  rapid  growth  of  radio  communication,  the  appliances 
and  methods  used  have  undergone  frequent  and  radical  changes. 
In  this  growth,  progress  has  been  made  largely  by  new  inventions 
and  by  the  use  of  greater  power,  and  comparatively  little  attention 
paid  to  refinements  of  measurement.  In  consequence  the  methods 
and  instruments  of  measurement  peculiar  to  radio  science  have 
developed  slowly  and  have  not  yet  been  carried  to  a  point  where 
they  are  as  accurate  or  as  well  standardized  as  other  electrical 
measurements. 

This  circular  presents  information  regarding  the  more  important 
instruments  and  measurements  actually  used  in  radio  work.  It  is 
hoped  that  the  treatment  will  be  of  interest  and  value  to  Govern- 
ment officers,  radio  engineers,  and  others,  notwithstanding  the 
subject  is  not  completely  covered.  Many  of  the  matters  dealt 
with  are  or  have  been  under  investigation  in  the  laboratories  of 
this  Bureau  and  are  not  treated  in  previously  existing  publications. 
No  attempt  is  made  in  this  circular  to  deal  with  the  operation  of 
apparatus  in  sending  and  receiving.  It  is  hoped  to  deal  with 
such  apparatus  in  a  future  circular.  The  present  circular  will  be 
revised  from  time  to  time,  in  order  to  supplement  the  information 
given  and  to  keep  pace  with  progress.  The  Bureau  will  greatly 
appreciate  suggestions  from  those  who  use  the  publication  for 
improvements  or  changes  which  would  make  it  more  useful  in 
military  or  other  service. 

The  methods,  formulas,  and  data  used  in  radio  work  can  not 
be  properly  understood  or  effectively  used  without  a  knowledge 
of  the  principles  on  which  they  are  based.  The  first  part  of  this 
circular,  therefore,  attempts  to  give  a  summary  of  these  principles 

5 


6  Circular  of  the  Bureau  of  Standards 

in  a  form  that' is  as  simple  as  is  consistent  with  accuracy.  A  large 
proportion  of  this  publication  is  devoted  to  the  treatment  of 
fundamental  principles  for  the  reasons,  first,  that  however  much 
the  methods  and  technique  of  radio  measurement  may  change  the 
same  principles  continue  to  apply,  and  second,  that  this  will  make 
the  present  circular  serve  better  as  an  introduction  to  other 
circulars  on  radio  subjects  which  may  be  issued. 

A  familiarity  with  elementary  electrical  theory  and  practice 
is  assumed.  Introductory  treatment  of  electrostatics  and  mag- 
netic poles,  electric  and  magnetic  fields,  the  laws  of  direct  currents, 
and  descriptions  of  the  more  common  electric  instruments  and 
experiments  may  be  found  in  many  books.  A  list  of  publications 
suitable  as  an  introduction  to  the  theory  given  in  this  circular 
may  be  found  in  the  bibliography  (p.  324).  The  common  explana- 
tion of  electric  current  as  similar  to  the  flow  of  water  in  a  pipe, 
while  adequate  for  most  of  the  phenomena  of  direct  current  is  not 
suitable  for  alternating  currents  and  particularly  for  radio.  The 
explanations  here  given  attempt  to  give  a  better  insight  into  the 
behavior  of  electric  current.  Most  of  the  treatment  of  principles 
is  a  presentation  of  the  theory  of  low-frequency  alternating 
currents,  arranged  with  its  radio  applications  in  mind.  There  is 
little  in  the  way  of  special  theory  before  section  24,  which  deals 
with  damped  waves,  and  yet  the  underlying  principles  of  the  chief 
radio  phenomena  are  covered.  Furthermore,  damped  waves  are 
of  less  importance  than  formerly,  since  modern  practice  tends 
toward  the  exclusive  use  of  continuous  or  undamped  waves.  The 
principles  of  radio  measurements  are  thus  nearly  identical  with 
those  of  any  other  alternating-current  measurements. 

THE  FUNDAMENTALS  OF  ELECTRO  MAGNETISM 
1.  ELECTRIC  CURRENT 

Electric  current  is  the  rate  of  flow  of  a  quantity  of  electricity. 
The  most  familiar  and  most  useful  properties  of  an  electric  cur- 
rent are  (i)  the  heating  effect  produced  in  a  conductor  in  which 
it  flows,  and  (2)  the  magnetic  field  surrounding  it.  The  latter  is 
by  far  the  most  important  property  in  radio  work.  The  study 
of  the  combined  effects  of  electricity  and  the  magnetism  accom- 
panying an  electric  current  constitutes  the  subject  of  electro- 
magnetism. 

When  a  current  flows  continuously  in  the  same  direction,  as 
the  current  from  a  battery,  it  is  called  a  direct  current.  When  the 


Radio  Instruments  and  Measurements  7 

current  periodically  reverses  in  direction,  it  is  an  alternating  cur- 
rent. The  alternation  of  current  is  accompanied  by  a  reversal 
of  direction  of  the  magnetic  field  around  the  current.  On  this 
account  alternating  currents  behave  very  differently  from  direct 
currents.  The  uses  of  alternating  currents  may  be  divided, 
roughly,  into  three  groups,  separated  according  to  the  frequency 
of  alternation  of  the  currents  used: 

Electric  power  applications,  20  to  100  per  second. 

Telephony,  100  to  20  ooo  per  second. 

Radio,  20  ooo  to  2  ooo  ooo  per  second. 

Displacement  Currents. — Direct  currents  can  flow  continuously 
in  conductors  only,  while  alternating  currents  flow  also  in  insula- 
tors. Suppose  a  circuit  contains  a  condenser,  consisting  of  two 
large  metal  plates  separated  by  air  or  some  other  insulating  me- 
dium. If  a  battery  is  connected  into  the  circuit,  a  momentary 
flow  of  current  takes  place,  accompanied  by  an  electric  strain  in  the 
insulating  medium  of  the  condenser.  This  strain  is  opposed  by 
an  electric  stress,  which  soon  stops  the  current  flow.  The  action 
is  much  like  the  flow  of  gas  under  pressure  into  a  gas  tank,  as 
described  below  in  the  section  on  capacity.  The  flow  of  gas  stops 
as  soon  as  the  back  pressure  of  the  compressed  gas  in  the  tank  is 
equal  to  the  applied  pressure.  The  flow  of  electric  current  stops 
as  soon  as  the  back  electric  pressure  (called  "  potential  difference  ") 
of  the  electrically  strained  medium  is  equal  to  the  electric  pressure 
("electromotive  force".)  of  the  battery. 

If  there  is  a  source  of  alternating  current  in  a  circuit  containing 
a  condenser,  the  electric  strain  in  the  insulating  medium  reverses 
in  direction  for  every  alternation  of  the  current.  The  electric 
strain  of  which  we  have  been  speaking  is  called  electric  displace- 
ment and  its  variation  gives  rise  to  a  so-called  displacement  cur- 
rent. The  electric  strain  is  of  two  kinds:  First,  there  is  the 
strain  in  the  actual  material  dielectric.  This  part  is  a  movement 
of  electricity  in  the  same  sense  as  the  transference  of  a  definite 
quantity  of  electricity  through  a  wire  is  a  movement  of  elec- 
tricity, the  only  difference  being  that  in  the  insulator  there  is  a 
force  (which  we  called  electric  stress)  which  acts  against  the 
electric  displacement.  Second,  there  is  the  strain  which  would 
exist  in  the  ether  if  the  material  dielectric  were  absent.  Some 
prefer  to  think  of  this  as  a  sort  of  electric  displacement  in  the 
ether,  of  a  kind  similar  to  that  in  the  matter;  but  a  knowledge  of 
the  physical  nature  of  electric  displacement  is  unnecessary  for 
practical  purposes ;  all  that  is  necessary  is  a  statement  of  how  the 


8  Circular  of  the  Bureau  of  Standards 

displacement  current  is  to  be  denned  as  measured,  and  of  the  rela- 
tion of  the  quantity  so  measured  to  other  electromagnetic 
quantities. 

The  effect  of  electric  displacement  at  any  point  in  a  medium  is 
handed  on  to  adjacent  points  and  so  spreads  out  through  space. 
Under  certain  conditions  a  considerable  quantity  of  this  moving 
displacement  and  the  magnetic  field  accompanying  it  become 
detached  from  the  circuit.  This  process  is  what  constitutes  the 
radiation  of  electromagnetic  waves,  which  makes  radio  commu- 
nication possible. 

Electrons. — The  flow  of  an  electric  current  in  a  conductor  is  not 
opposed  by  electric  stress  as  in  an  insulator.  A  current  in  a  con- 
ductor is  believed  to  consist  of  the  motion  of  immense  numbers 
of  extremely  small  particles  of  electricity,  called  electrons.  All 
electrons  are,  so  far  as  known,  strictly  identical,  are,  for  ease  of 
calculation,  assumed  to  be  spherical  in  shape,  and  have  the  fol- 
lowing dimensions,  etc.:  Radius,  i  X  io~13  centimeter;  mass,  8.8X 
io~28  gram;  electric  charge,  1.59  x  io~19  coulomb. 

An  electron  is  thousands  of  times  smaller  than  any  atom. 
Views  as  to  the  transference  of  the  electric  current  by  the  motion 
of  electrons  in  a  conductor  have  undergone  considerable  changes 
during  the  last  few  years.  Some  of  the  electrons  in  a  conductor 
are  bound  to  the  molecules  while  others  are  free  to  move  about. 
The  latter  are  in  constant  motion  between  the  molecules,  in  zigzag 
paths  because  of  repeated  collisions  with  the  molecules.  The 
motion  of  the  free  electrons  is  thus  very  similar  to  the  heat  mo- 
tions of  the  molecules;  in  fact,  the  average  kinetic  energy  of  an 
electron  is  equal  to  that  of  a  molecule  at  the  same  temperature. 
The  average  velocity  of  the  electrons  is  about  100  km  per  second 
at  o°  C.  This  increases  with  temperature.  Until  recently  it  was 
supposed  that  conduction  in  a  solid  takes  place  almost  entirely 
through  the  agency  of  the  free  electrons.  The  electric  current 
was  nothing  more  than  a  slow  drift  (a  sort  of  electric  wind)  super- 
posed upon  the  random  motions  of  the  electrons  by  the  electric 
field.  This  view,  although  suggestive  and  fruitful,  is  attended 
with  many  difficulties,  and  the  present  tendency  is  toward  some 
form  of  theory  in  which  the  conduction  is  brought  about  by  the 
spontaneous  discharge  of  electrons  from  one  molecule  to  another, 
the  function  of  the  field  being  to  influence  the  orientation  of  the 
discharge,  which  would,  in  its  absence,  be  perfectly  random. 
The  study  of  electrons  has  recently  led  to  great  improvements  in 


Radio  Instruments  and  Measurements  9 

apparatus  for  producing  and  detecting  currents  of  radio  frequency. 
(See  sec.  56.) 

When  electricity  is  in  equilibrium  in  a  conductor,  the  electric 
charge  is  in  a  very  thin  layer  upon  its  surface;  thus  the  phenom- 
ena of  electrostatics  arise.  An  insulator  is  believed  to  contain 
no  free  electrons.  The  electrons  are  bound  to  the  molecules  in 
such  a  way  that  they  can  be  slightly  displaced  by  an  electric  force 
but  return  to  their  positions  of  equilibrium  when  it  is  removed. 
This  motion  of  the  bound  electrons,  together  with  the  electric 
strain  in  the  ether  itself,  constitutes  the  electric  displacement  in 
the  insulator,  and  determines  its  dielectric  constant.  Another 
type  of  motion  of  which  the  electrons  attached  to  molecules  are 
capable  is  vibration  about  their  positions  of  equilibrium.  They 
thus  give  rise  to  waves  that  travel  outward  in  the  form  of  light 
and  heat  radiation. 

2.  ENERGY 

Most  useful  operations  in  physics  involve  the  movement  of 
something  from  one  place  to  another;  and  in  general,  as  for 
example  when  the  body  moved  is  held  by  a  spring,  or  when  its 
velocity  changes  during  the  motion,  force  has  to  be  exerted  to 
cause  the  motion.  A  useful  quantity  which  figures  in  the  discus- 
sion of  such  motions  is  the  quantity  called  "work."  The  work 
done  upon  a  body  is  defined  as  the  product  of  the  force  which 
acts  upon  the  body  into  the  distance  moved;  or  (when  the  force 
varies  during  the  motion)  as  the  sum  of  such  products  for  each 
element  of  path  described  by  the  body.  When  a  body  or  system 
of  any  kind  possesses  .the  power  to  do  work  in  virtue  of  its  posi- 
tion, velocity,  chemical  constitution,  or  any  other  feature,  it  is 
said  to  possess  energy;  and  the  measure  of  the  change  of  energy 
which  it  experiences  in  doing  such  work  is  the  amount  of  work 
done.  One  of  the  fundamental  principles  of  science  is  the  "con- 
servation of  energy."  The  amount  of  energy  in  existence  is  con- 
stant; energy  can  not  be  created  nor  destroyed;  it  can  only  be 
transformed  from  one  form  into  another.  It  is  often  very  helpful 
to  the  understanding  of  a  process  to  consider  what  energy  changes 
are  taking  place.  The  transformation  from  one  form  into  another 
is  always  accompanied  by  a  dissipation  of  some  of  the  energy  as 
heat  or  some  other  form  in  which  it  is  no  longer  available  for  the 
use  desired.  Thus  while  none  of  the  energy  is  lost  during  an 
energy  change,  more  or  less  of  it  becomes  no  longer  available. 


io  Circular  of  the  Bureau  of  Standards 

Kinds  of  Energy. — The  familiar  kinds  of  energy  are  mechanical 
energy,  heat,  chemical  energy  and  electrical  energy.  To  these 
may  be  added  radiant  energy,  but  this  is  considered  to  be  a  form 
of  electrical  energy.  Mechanical  energy  is  of  two  kinds,  kinetic 
energy  and  potential  energy.  When  an  object  is  in  motion  it  is 
said  to  possess  kinetic  energy.  If  the  motion  is  stopped  the 
kinetic  energy  of  the  object  changes  into  some  other  form.  For 
example,  if  the  moving  object  is  stopped  by  suddenly  striking  an 
immovable  obstacle  its  kinetic  energy  is  converted  into  heat.  If 
when  stopping  it  starts  another  object  moving,  the  second  object 
then  has  kinetic  energy.  Any  object  of  mass  ra  moving  with 

velocity  v  has  a  kinetic  energy  =  -  m  i;2. 

The  energy  which  an  object  possesses  in  virtue  of  its  position  is 
called  potential  energy.  A  stone,  lifted  a  certain  distance  above 
the  earth,  will  fall  if  released.  It  then  acquires  kinetic  energy  in 
falling.  It  had  a  certain  amount  of  potential  energy  when  at  the 
highest  point,  simply  in  virtue  of  its  position  above  the  earth. 
As  it  falls  this  potential  energy  is  being  changed  into  kinetic 
energy,  and  when  it  is  just  about  to  strike  the  ground  the  potential 
energy  has  all  been  converted  into  kinetic.  This  is  a  simple  exam- 
ple of  the  principle  of  conservation  of  energy. 

When  a  change  of  energy  from  one  form  into  another  -occurs, 
work  is  done.  When  an  object  falls  to  the  earth  from  a  height 
there  is  a  change  of  potential  energy  into  kinetic  and  work  is 
done  upon  the  object  by  the  force  of  gravity,  the  amount  of 
which  is  equal  to  the  product  of  the  force  by  the  distance  through 
which  the  object  falls.  Again,  when  a  body  is  moved  against  a 
force  tending  to  oppose  the  motion,  work  must  be  done  by  the 
agency  which  moves  the  body.  The  product  of  these  two  factors, 
the  force  acting  and  the  displacement  of  the  object,  is  the  amount 
of  work  done  in  an  energy  transformation. 

Electrical  Energy. — There  are  two  kinds  of  electrical  energy, 
similar  to  the  two  kinds  of  mechanical  energy.  Corresponding  to 
potential  energy  there  is  electrostatic  energy,  which  is  the  energy 
of  position  of  electricity  at  rest;  this  is  the  form  in  which  electric- 
ity is  stored  in  a  charged  condenser.  Corresponding  to  kinetic 
energy  there  is  electrokinetic  energy  (also  called  magnetic  energy) , 
which  is  the  energy  of  electricity  in  motion.  The  latter  is  the 
energy  of  the  electric  current,  and  is  associated  with  the  mag- 
netic field  accompanying  the  current.  In  accordance  with  the 
law  of  conservation  of  energy  the  sum  of  the  electrostatic  and 


Radio  Instruments  and  Measurements  1 1 

the  magnetic  energies  in  any  electrical  system  is  constant  if  the 
system  as  a  whole  does  not  receive  or  give  out  any  energy,  or  if 
energy  is  being  supplied  at  the  same  rate  at  which  it  is  being 
dissipated. 

Electrical  energy  can  readily  be  converted  into  other  types  of 
energy ;  if  this  were  not  so  it  would  not  be  the  important  factor 
in  the  life  of  man  that  it  now  is.  As  far  as  radio  science  is  con- 
cerned, the  two  principal  forms  of  energy  into  which  electrical 
energy  is  transformed  are  heat  and  electromagnetic  radiation. 
From  any  electrical  system  there  is  a  continuous  dissipation  or 
loss  of  electrical  energy  going  on,  and  in  general  the  evolution  of 
heat  in  the  circuit  plus  the  energy  radiated  as  electromagnetic 
waves  equals  the  diminution  of  the  sum  of  the  electrostatic  and 
magnetic  energies.  The  energy  of  electromagnetic  waves  is  a 
form  of  electric  energy,  being  a  combination  of  electrostatic  and 
magnetic  energies.  Inasmuch,  however,  as  it  travels  through 
space  entirely  detached  from  the  sending  circuit,  it  represents  a 
loss  of  energy  from  that  circuit. 

3.  RESISTANCE 

The  dissipation  or  loss  of  electrical  energy  is  expressible  in  terms 
of  resistance.  The  rate  of  evolution  of  energy  at  any  instant  in 
a  conductor  is  the  product  of  the  electromotive  force  acting  in 
the  conductor  by  the  current  flowing.  This  energy  usually  mani- 
fests itself  in  the  form  of  heat.  The  time  rate  of  energy  is  called 
power.  Thus, 

w        •     E>  -,     & 

p  =  -  =  e  i  =  R  i"  =  -p 

t  j\. 

£ 

Resistance  is  defined  by  R  =  -.     Power  (p)  is  generally  expressed 

i 

in  watts,  energy  (w)  in  joules,  electromotive  force  (e)  in  volts, 
current  (i)  in  amperes,  and  resistance  (R)  in  ohms,  unless  other- 
wise stated.  The  resistance  of  a  conductor  depends  on  the  mate- 
rial of  which  it  is  made,  the  size  and  shape  of  the  conductor,  and 
the  frequency  of  the  current.  The  characteristic  property  of  the 
material  is  called  its  resistivity.  Denoting  by  p  the  ordinary  or 
volume  resistivity,  by  /  the  length,  by  S  the  cross  section  of  the 
conductor,  and  by  R0  the  resistance  to  direct  current, 

^0  =  P5 

The  resistance  of  a  system  of  conductors  is  readily  found  by  the 
simple  laws  of  series  and  parallel  combination,  for  direct  currents. 


1 2  Circular  of  the  Bureau  of  Standards 

With  alternating  currents,  however,  the  calculation  is  more  diffi- 
cult, and  it  is  usually  found  convenient  to  utilize  the  relation 

*-£  <» 

The  resistance  of  a  single  conductor  to  alternating  currents  is 
found  by  the  aid  of  the  same  relation.  It  may  be  shown  that 
the  distribution  of  direct  current  in  a  system  of  conductors  or 
over  the  cross  section  of  a  single  conductor  is  such  as  to  make 
the  production  of  heat  a  minimum ;  and  it  results  that  in  a  single 
uniform  conductor  the  current  is  uniformly  distributed  over  the 
cross  section.  When  alternating  current  flows  in  a  conductor,  it 
tends  to  flow  more  in  the  outer  portions  of  the  conductor  than 
in  the  center.  In  consequence  of  this  change  of  current  distribu- 
tion, the  power  which  is  converted  into  heat  increases.  The 
higher  the  frequency  the  farther  does  the  current  distribution 
depart  from  the  direct-current  distribution,  and  the  greater  does 
the  power  p  become.  It  follows,  in  accordance  with  equation  (i), 
that  the  resistance  increases  as  frequency  increases. 

Radio-Frequency  Resistance. — With  alternating  currents  the 
departure  from  uniform  distribution  of  the  current  is  spoken  of 
as  the  skin  effect.  At  high  frequencies  the  current  flows  in  a  thin 
layer  at  the  surfaces  of  conductors,  and  the  skin  effect  is  thus  large 
in  all  except  very  thin  wires;  the  resistances  of  ordinary  con- 
ductors at  radio  frequencies  may  be  many  times  their  low-fre- 
quency resistances.  The  ratio  of  resistance  at  any  frequency  to 
the  direct-current  or  low-frequency  resistance  can  be  calculated 
for  certain  simple  forms  of  conductors.  Formulas  for  this  are 
given  below  in  sections  74  to  76.  In  most  practical  cases,  how- 
ever, the  radio  resistance  can  be  obtained  only  by  measurement. 

In  addition  to  the  resistances  of  conductors,  resistance  is  intro- 
duced into  radio  circuits  by  three  other  causes,  viz,  sparks,  dielec- 
trics, and  radiation.  Dielectric  resistance  is  treated  in  section  34, 
below.  The  energy  radiated  from  a  circuit  per  unit  time  in  elec- 
tromagnetic waves  is  proportional  to  the  square  of  the  current  in 
the  circuit.  It  is  thus  analogous  to  the  energy  dissipation  as  heat 
in  a  conductor,  and,  therefore,  the  radiation  increases  the  effective 
or  equivalent  resistance  by  a  certain  amount.  This  added  resist- 
ance is  conveniently  called  radiation  resistance.  It  can  be  cal- 
culated for  a  few  simple  types  of  circuit.  It  is,  in  general,  large 
enough  to  be  appreciable  only  when  the  circuit  has  the  open  or 
antenna  form,  or  when  a  closed  circuit  is  of  large  area  and  the 
frequency  is  high. 


Radio  Instruments  and  Measurements  13 

4o  CAPACITY 

Electrostatic  energy  may  be  stored  in  an  arrangement  of  con- 
ductors and  insulator  called  an  electrical  condenser.  The  action 
of  a  condenser  is  somewhat  similar  to  that  of  a  gas  tank  used  for 
the  storage  of  gas.  The  amount  of  gas  a  tank  will  hold  is  not  a 
constant,  fixed  amount;  it  depends  on  the  pressure.  If  the  pres- 
sure is  doubled,  twice  the  mass  of  gas  is  forced  into  the  tank.  The 
internal  or  back  pressure  of  the  gas  opposes  the  applied  pressure. 
If  the  applied  pressure  is  released  and  an  opening  is  left  in  the 
tank,  the  gas  rushes  forth. 

The  amount  of  electric  charge  given  to  a  condenser  depends  on 
the  electric  pressure,  or  potential  difference;  and  in  exact  simi- 
larity to  the  gas,  the  charge  is  proportional  to  this  potential  dif- 
ference. The  constant  ratio  of  charge  to  potential  difference  is 

called  the  capacity  of  the  condenser.     In  symbols,  ^  =  C,     The 

capacity  of  a  condenser  depends  on  the  size  and  distance  apart  of 
its  plates,  and  on  the  kind  of  dielectric  between  the  plates.  (Vari- 
ous kinds  of  condensers  are  described  in  sec.  32,  etc.,  and  formu- 
las for  calculating  capacity  are  given  in  sees.  63  to  65.)  The 
applied  potential  difference  is  opposed  by  a  sort  of  elastic  re- 
action of  the  electricity  in  the  condenser,  just  as  the  internal 
pressure  of  the  gas  in  a  tank  opposes  the  external  applied  pressure. 
If  the  plates  of  a  charged  condenser  are  connected  by  a  conductor, 
with  no  applied  electromotive  force,  the  condenser  discharges. 

The  Dielectric. — The  insulating  medium  in  a  condenser  is  called 
a  dielectric.  The  process  of  charging  causes  electric  displacement 
in  the  dielectric.  When  a  body  is  moved  against  a  force  tending 
to  prevent  the  motion,  work  is  done,  and  similarly,  when  a  con- 
denser is  charged  against  the  quasi-elastic  reaction  of  the  dielectric, 
work  is  done  upon  the  condenser.  The  energy  of  the  charging 
source  is  stored  up  as  electrostatic  energy  in  the  dielectric.  The 
two  factors  upon  which  the  energy  depends  are  the  charge  and  the 
potential  difference, 


Since  p.  =  C,  we  have  also 

~~2 


14  Circular  of  the  Bureau  of  Standards 

The  pressure  upon  the  gas  in  a  tank  can  not  be  increased 
indefinitely,  for  the  tank  will  ultimately  yield  and  break.  Simi- 
larly there  is  a  limit  to  the  potential  difference  which  can  be 
applied  to  a  condenser,  for  the  dielectric  will  be  broken  down,  or 
punctured,  if  the  limit  is  exceeded.  The  potential  difference  at 
which  a  spark  will  pass  and  the  dielectric  be  punctured  is  called 
the  "  dielectric  strength." 

Capacity  is  one  of  the  two  quantities  of  chief  importance  in 
radio  circuits.  The  other  is  inductance,  treated  in  the  following 
section. 

5,  INDUCTANCE 

Magnetic  Flux. — The  physical  quantity  called  inductance  is 
dependent  upon  the  magnetic  field  which  surrounds  every  electric 

current.  The  intensity  of  this 
magnetic  field  at  any  point  is 
proportional  to  the  current. 
The  direction  of  the  magnetic 
field  around  a  straight  wire  car- 
rying a  current  is  given  by  Am- 
pere's right-hand  rule :  Close  the 
right  hand  with  the  thumb  ex- 

FIG.  i.— Direction  of  magnetic  field  around    tended;  point  the  thumb  in  the 
a  wire  carrying  a  current  direction  of  the  Current  flow ;  the 

magnetic  field  is  then  in  the  direction  in  which  the  fingers  point, 
in  circles  in  planes  perpendicular  to  the  wire.  The  magnitude  of 
the  magnetic  field  intensity  can  be  easily  computed  for  some  sim- 
ple forms  of  circuit  from  the  principle  that  its  line  integral *  in  a 

path  completely  around  the  current  is  equal  to  :  -  times  the  cur- 
rent in  amperes. 

As  an  example,   suppose  a  current  i  flowing  in  a  very  long 
solenoid  of  N  turns,  radius  r  and  length  /.     The  magnetic  field 
intensity  H  is  parallel  to  the 
axis  and  of  constant  value  in- 
side the  solenoid;  it  may  be 
shown  to  be  zero  in  the  space 
outside,  and  the  effects  of  the 
ends  may  be  neglected.     The  WT^        c.    .,      ,     ., 

J  TIG.  2. — simple  solenoid 

line  integral  of  H  along  any 

path  completely  around  the  current   is  HI  inside  the  solenoid 

1  The  line  integral  of  a  quantity  along  any  line  or  path  is  the  sum  of  the  products  of  the  length  of  each 
element  of  the  path  by  the  value  of  the  quantity  along  that  element.  If  the  quantity  has  a  constant  value 
along  the  whole  path,  the  line  integral  is  simply  the  product  of  this  value  by  the  length  of  the  path. 


Radio  Instruments  and  Measurements 


and  is  zero  for  the  rest  of  the  path. 
i,  N  times.     Hence 


The  path  incloses  the  current 


, 
10 

4£  Ni 
10    / 


(2) 


The  magnetic  field  in  the  medium  surrounding  a  conductor 
carrying  a  current  produces  a  magnetized  condition  of  the  medium. 
This  condition  is  a  sort  of  magnetic  strain  in  the  medium  and  is 
analogous  to  displacement  produced  in  a  dielectric  by  electric 
potential  difference.  The  amount  of  this  magnetic  strain  through 
any  area  is  called  the  magnetic  flux.  This  quantity  (for  which 
the  symbol  <j>  is  used)  is  equal  to  the  product  of  the  three  factors, 
magnetic  field  intensity,  area  and  the  magnetic  permeability. 
Permeability  is  a  property  of  matter  or  of  any  medium  which 
indicates,  so  to  speak, 
i  t  s  magnetizability. 
Its  numerical  value 
is  equal  to  unity  for 
empty  space  and  for 
air  and  most  sub- 
stances. Iron  may 
have  a  permeability 
as  high  as  10  ooo  or 
even  more. 

Self  -  inductance. — 
Inductance  is  a  quan- 
tity introduced  as  a 
convenient  means  of  dealing  with  magnetic  fluxes  associated  with 
currents.  The  self-inductance  of  a  circuit  is  simply  the  total  mag- 
netic flux  linked  with  the  circuit  due  to  a  current  in  the  circuit, 
per  unit  of  current.  In  symbols, 

d>  f   •. 

L-?  (3) 

The  magnitude  of  L  depends  on  the  shape  and  size  of  the  circuit 
and  is  a  constant  for  a  given  circuit,  the  surrounding  medium 
being  of  constant  permeability.  If  the  circuit  has  N  turns  each 
traversed  by  the  same  magnetic  flux  <j>,  then  when  L  is  expressed 
in  terms  of  the  usual  unit,  called  the  "  henry," 

N<f>  ,  \ 

L  =  — —  (4) 


FlG"  3  •— 


lux  around  a  solenoid  in  which  a  current 
is  flowing. 


io 


35601 c 


i6 


Circular  of  the  Bureau  of  Standards 


Analogy  of  Inductance  to  Inertia. — The  magnetic  fluxx  asso- 
ciated with  a  current  is  analogous  to  the  momentum  associated 
with  a  moving  body.  Because  of  its  inertia  or  mass  (m) ,  a  body 
in  motion  with  velocity  (v)  opposes  any  change  in  its  momentum 
(mv).  Inertia  is  obviously  a  very  different  thing  from  friction, 
which  always  resists  the  motion  and  tends  to  decrease  the  velocity 
or  momentum.  Inertia  only  opposes  a  change  in  momentum, 
\J  "V  ano^  hence  does  not  affect  a 

motion  with  constant  velocity. 
Inductance  may  be  spoken  of 
as  electrical  inertia  or  mass. 
The  inductance  (L)  of  a  cir- 
cuit in  which  a  current  (i)  is 

FIG.  4.— A  body  having  a  mass  (m)  and  a  velocity  flowing  Opposes  any  change  in 
(v)  opposes  any  change  in  its  momentum  (mv)  ^he  flux  (d>) .  Electrical  resist- 
ance and  inductance  are  very  different,  for  resistance  behaves  like 
mechanical  friction,  opposing  even  a  constant  current,  while  in- 
ductance only  opposes  a  change  in  the  current.  Thus  inductance 
has  no  effect  on  constant  direct  currents  but  is  one  of  the  deter- 
mining factors  in  the  flow  of  alternating  currents.  The  analogy 
of  inductance  to  inertia,  of  current  to  velocity,  and  of  flux  to 
momentum,  will  be  brought  out  further  in  the  next  section. 

Mutual  Inductance. — A  part  of  the  magnetic  flux  from  a  circuit 
may  pass  through  or  link  with  a  second  circuit.     The  amount  of 


FIG.  5. — Linking  of  magnetic  flux  of  one  circuit  with  another;  the  basis  oj 
the  conception  of  mutual  inductance 

this  flux  linked  with  circuit  2,  per  unit  of  current  in  circuit  i,  is 
called  the  mutual  inductance  of  the  two  circuits.  If  <£12  denotes 
the  flux  mentioned  and  %  =  current  in  circuit  i ,  the  mutual 
inductance  = 


Radio  Instruments  and  Measurements  1  7 

It  is  also  true  that  if  <£21=the  flux  from  circuit  2  linked  with 
circuit  i  ,  and  i2  =  current  in  circuit  2, 


The  magnitude  of  any  mutual  inductance  depends  on  the  shape 
and  size  of  the  two  circuits,  their  positions  and  distance  apart, 
and  the  permeability  of  the  medium.  If  there  are  N^  turns  in 
the  first  circuit  and  N2  turns  in  the  second  and  the  same  amount 
of  flux  from  one  passes  through  every  turn  of  the  other,  then  using 
the  <£  's  to  denote  that  part  of  the  flux  from  one  turn  of  either 
circuit  which  passes  through  each  turn  of  the  other  circuit,  and 
using  the  usual  units, 


If,  however,  the  <£  's  denote  the  total  flux  from  either  circuit  pass- 
ing through  each  turn  of  the  other  circuit,  this  becomes 

M=^  (5) 

10% 

For  any  of  the  definitions  of  0,  the  ratio  -^  =  -^,  and  is  a  quan- 

i2      i1 

tity  depending  only  on  geometrical  configuration. 

Calculation  of  Inductance. — It  is  frequently  convenient  to  deal 
with  the  self-inductance  of  a  particular  coil  or  with  the  mutual 
inductance  of  limited  portions  of  two  circuits.  Inductance  is 
strictly  defined  only  for  complete  circuits.  The  self-inductance 
of  a  part  of  a  circuit  is  understood  to  be  such  that  the  inductance 
of  the  complete  circuit  is  equal  to  the  sum  of  the  self -inductances 
of  all  the  parts  and  the  mutual  inductance  of  every  part  with 
every  other  part. 

Inductances  are  computed  by  the  aid  of  equation  (3) ,  together 
with  the  principle  given  above  that  flux  is  field  intensity  times 
area  times  permeability.  For  example,  to  find  the  inductance  of 
the  long  solenoid  of  Fig.  2, 


1  8  Circular  of  the  Bureau  of  Standards 

in  which  5  =  the  area  of  the  circular  cross  section  of  the'  solenoid 
and  jot  =  i  ,  the  permeability  of  air.  Substituting  the  value  of  H 
from  (2)  ,  and  putting  5"  =  irr2, 

_  N      4-n-Ni  _47T27VV2 

•  X          j~  X  irr  X  I 


-f  —      o  •  j  —      o     T 

10sl       10    /  I09      / 

Change  of  Inductance  with  Frequency.  —  When  it  is  desired  to 
calculate  inductance  with  great  accuracy,  account  must  be  taken 
of  the  magnetic  flux  within  the  conductor  carrying  the  current, 
as  well  as  the  flux  outside  the  conductor.  The  flux  in  a  wire  is 
greatest  at  the  circumference  and  zero  at  the  axis  of  the  wire, 
because  the  flux  is  due  only  to  the  current  which  it  surrounds. 
Any  change  in  the  distribution  of  current  within  the  wire  changes 
also  the  flux  distribution  and  hence  the  inductance.  As  has 
been  stated  in  the  section  on  resistance,  the  current  distribution 
is  different  for  different  frequencies.  Consequently  inductance 
varies  with  frequency.  As  the  frequency  is  increased,  less  cur- 
rent flows  near  the  axis  of  the  wire  and  more  flows  in  the  surface 
portions.  The  flux  in  the  central  parts  of  the  wire  is  thus  dimin- 
ished, and  the  inductance  decreases  as  frequency  increases.  This 
effect  is  small,  because  the  whole  flux  within  the  wire  is  a  small 
part  of  the  total  flux.  (See  formulas  (131)  to  (138)  in  sec.  67.) 
There  is  a  similar  change  of  mutual  inductance  with  frequency, 
but  it  is  so  small  as  to  be  wholly  negligible. 

Series  and  Parallel  Arrangement  of  Inductances.  —  Inductances  in 
series  add  like  resistances.  When  the  coils  or  conductors  which 
are  combined  are  so  far  apart  that  mutual  inductances  are  negli- 
gible, inductances  in  parallel  are  combined  like  resistances  in 
parallel.  Taking  account  of  mutual  inductance,  the  total  induc- 
tance of  any  number  of  inductances  in  series  is 


Some  or  all  of  the  mutual  inductances  may  be  negative.     For 
two  coils  in  parallel,  the  total  inductance  is 


L1+L2-2M 


The  last  term  in  the  denominator  changes  sign  if  the  coils  are  so 
connected  that  M  is  negative.  This  expression  applies  at  radio 
frequencies,  but  at  low  frequencies  the  resistance  of  the  coil  may 
have  to  be  taken  into  account.  For  more  than  two  inductances 
in  parallel,  the  expression  for  the  total  inductance  is  complicated. 


Radio  Instruments  and  Measurements  19 

THE  PRINCIPLES  OF  ALTERNATING  CURRENTS 
6.  INDUCED  ELECTROMOTIVE  FORCE 

When  the  magnetic  flux  through  any  circuit  is  changing,  an 
electromotive  force  is  produced  around  the  circuit,  which  lasts 
while  the  change  is  going  on.  The  change  of  flux  may  be  caused 
in  various  ways;  a  magnet  may  be  moved  in  the  vicinity,  the 
circuit  or  a  part  of  it  may  be  moved  while  near  a  magnet,  the  cur- 
rent in  a  second  near-by  circuit  may  be  altered,  or  either  circuit 
or  a  part  thereof  may  be  moved.  The  electromotive  force  thus 
caused  is  called  an  induced  electromotive  force,  and  the  result- 
ing current  in  the  circuit  is  an  induced  current.  The  direction  of 
the  induced  emf  and  current  is  given  by  L/enz's  law,  viz,  an 
induced  current  always  flows  in  such  a  direction  as  to  oppose 
the  action  which  produces  it.  For  example,  if  the  current  is 
induced  by  bringing  a  magnet  near  a  circuit,  the  current  in  the 
circuit  will  be  such  as  to  repel  the  magnet.  The  energy  of  the 
induced  current  results  from  the  work  necessary  to  bring  up  the 
magnet  against  the  repelling  force. 

The  magnitude  of  the  induced  emf  at  any  instant  is  in  every 
case  equal  to  the  rate  of  change  of  the  magnetic  flux  through  the 
circuit.  This  is  expressed  by  the  formula 


where  e  represents  the  instantaneous  value  of  the  induced  emf 
in  a  circuit  consisting  of  a  simple  loop  or  single  turn,  and  -~  is  an 

expression  called  the  derivative  2  of  flux  with  respect  to  time 
and  which  tells  the  instantaneous  rate  of  change  of  the  flux. 

This  quantity  -37  is  the  change  of  flux  during  a  very  small  inter- 

val of  time  divided  by  the  time,  and  its  value  may  vary  from 
instant  to  instant.  If  it  remains  constant  for  a  certain  length 
of  time,  then  its  value  is  the  whole  change  of  flux  during  that 
interval  divided  by  the  interval. 

If  the  changing  flux  through  the  circuit  is  the  flux  021  from  a 

second  circuit,  e  =  —jr-     This  flux  is  expressible  in  terms  of  mu- 

2  While  derivatives  are  used  in  a  few  places  in  this  circular,  it  is  believed  that  the  treatment  can,  never- 
theless, be  understood  by  a  person  not  familiar  with  calculus.  To  avoid  the  use  of  derivatives  entirely 
would  require  circumlocution  such  that  the  treatment  would  doubtless  be  even  less  clear. 


2O  Circular  of  the  Bureau  of  Standards 

tual  inductance  and  the  current  in  the  second  circuit,  thus,  4>2l  =  Mi2. 

Consequently, 

_d(Mii). 
dt 

If  the  circuits  are  fixed  in  position,  M  remains  constant,  so  this 
becomes 

e  =  M-^  (7) 

Thus  the  electromotive  force  induced  in  a  circuit  by  variation 
of  current  in  another  circuit  is  equal  to  the  product  of  the  mutual 
inductance  by  the  rate  of  change  of  current  in  the  second  circuit. 
An  emf  may  also  be  induced  in  one  circuit  by  a  variation  of  the 
current  in  the  circuit  itself.  Since  the  flux  associated  with  a 
current  is  tf>=Li,  it  follows  that  the  self -induced  emf  is  given  by 


That  is,  the  emf  induced  in  the  circuit  is  equal  to  the  product  of 
the  self -inductance  by  the  rate  of  change  of  the  current. 

When  the  flux  $  through  a  coil  of  N  turns  is  changing,  the  total 
emf  e.  induced  in  the  whole  coil  is  N  times  that  induced  in  one 
turn.  The  simple  equation  (6)  becomes,  in  terms  of  the  usual 
units, 

N  d*  ,  . 

e  =  ^~dt  (9) 

Equations  (7)  and  (8)  are  correct  when  emf  is  expressed  in  volts, 
inductance  in  henries,  current  in  amperes,  and  time  in  seconds. 
They  were  obtained  from  the  simpler  equation  (6)  for  induced 
emf,  but  they  need  no  modification  on  this  account,  because  equa- 
tions (4)  and  (5)  for  self  and  mutual  inductance  also  contain  the 
factors  N  and  io8  which  cancel  those  in  (9). 

Mechanical  Analog. — The  fact  that  a  change  of  magnetic  flux 
gives  rise  to  an  electromotive  force  which  opposes  the  change 
may  be  understood  by  recalling  that  a  change  of  mechanical 
momentum  of  a  body  is  opposed  by  the  force  of  inertia.  This 
force  is  equal  to  the  rate  of  change  of  momentum.  The  elec- 
trical and  mechanical  cases  are  strictly  analogous.  Flux  corre- 
sponds to  momentum,  electromotive  force  to  mechanical  force 
(F),  current  to  velocity  (u),  and  inductance  to  mass  (m).  If 
the  mass  is  constant,  we  have 

~       dv 


Radio  Instruments  and  Measurements 


21 


as  the  expression  that  force  equals  rate  of  change  of  momentum, 
analogous  to 

di 


7.  SINE  WAVE 

The  ordinary  dynamo  is  the  most  familiar  application  of  the 
principle  of  induced  emf.  The  field  magnets  give  rise  to  mag- 
netic flux,  and 
coils  of  wire 
(constituting  the 
armature)  are 
caused  to  move 
across  this  flux 
by  some  outside 
source  of  power. 

The  simplest 
type  of  dynamo 
generates  an  al- 
ternating current 


FIG.  6. — Simple  dynamo  illustrating  how  the  revolving  con- 
ductor cuts  magnetic  lines 


R 


and  is  sketched  in  Fig.  6.  The  single  turn  of  wire  shown  is  in 
such  a  position  that  maximum  magnetic  flux  passes  through  it. 
When  it  is  rotated  in  either  direction,  the  flux  passing  through  it 
is  changed,  and  hence  an  electromotive  force  is  induced  in  it.  If 

the  turn  of  wire  is  continuously  rotated 
at  constant  angular  speed  w,  the  rate  of 
change  of  magnetic  flux  through  it  will  be 
greater  in  some  positions  than  in  others, 
and  the  electromotive  force  at  the  slip- 
rings  A  A  will  vary  in  a  certain  manner, 
this  variation  being  repeated  each  revo- 
lution. In  Fig.  7  let  POP  represent  the 
end  view  of  the  turn  of  wire.  As  the 
wire  revolves  to  the  successive  positions 
PiPPi,  P2OP2,  etc.,  the  emf  is  propor- 
tional to  the  sine  of  the  angle  formed  by 
the  revolving  wire  with  POP.  The  mag- 
nitude of  the  emf  at  any  instant  may, 
therefore,  be  represented  by  the  vertical  lines  P^M^  P2M2,  etc., 
drawn  from  the  horizontal  axis  to  the  end  of  a  line  revolving  with 
the  angular  velocity  co. 

A  diagram  may  be  drawn,  taking  the  distance  along  a  hori- 
zontal line  to   represent   time   and   the  vertical  distance   from 


\/ 


FIG.  7. — Successive  positions 
of  revolving  conductor;  the 
emf  generated  is  propor- 
tional to  the  sine  of  the  angle 
formed  by  the  revolving  con- 
ductor and  POP 


22 


Circular  of  the  Bureau  of  Standards 


this  line  to  represent  induced  electromotive  force.  This  emf 
curve  has  the  mathematical  form  of  a  sine  wave.  Many  dyna- 
mos in  actual  use  generate  electromotive  forces  very  nearly  of 
this  form,  and  on  account  of  its  mathematical  simplicity  the  sine 
wave  is  assumed  in  most  of  alternating-current  theory.  It  should 
not  be  forgotten,  however,  that  sine-wave  theory  is  in  many  prac- 
tical cases  only  an  approximation  because  the  emf  is  not  rigor- 
ously of  sine-wave  form. 

Letting  e  =  emf  at  any  instant,  E0  =  maximum  emf  (that  at  the 
crest  of  the  wave  as  shown  in  Fig.  8) ,  t  =  time,  co  =  angular  velocity 
of  the  turn  of  wire  in  Fig.  6, 


=  E0  sin 


(10) 


This  emf  alternates  in  direction.     Starting  at  a,  Fig.  8,  it  passes 
through  a  set  of  positive  values,  then  through  a  set  of  negative 


0 


\ 


,'P 


Time 


FIG.  8. — Sine  wave  developed  from  circle  diagram 

values,  and  at  b  begins  to  repeat  the  same  "cycle. "  In  the  time 
of  one  complete  cycle,  represented  by  the  distance  a  b,  the  revolv- 
ing radius  OP  makes  one  complete  revolution,  or  passes  through 
the  angle  2?r  radians.  The  time  required  for  one  complete  cycle 
being  represented  by  T,  it  follows  that 

27T 


The  time  T  is  called  the  "period"  of  the  alternation.  It  is  the 
reciprocal  of  the  "frequency,"  which  is  the  number  of  times  per 
second  that  the  electromotive  force  passes  through  a  complete 
cycle  of  values.  It  follows,  denoting  frequency  by  /,  that 


(H) 


Radio  Instruments  and  Measurements  23 

In  considering  the  effect  of  frequency  in  electrical  phenomena  the 
quantity  w  is  found  more  convenient  and  appears  oftener  than  /. 

Effective  Values  of  Alternating  Quantities. — The  instantaneous 
rate  at  which  heat  is  produced  in  a  circuit  is  proportional  to  the 
square  of  the  instantaneous  current.  According  to  the  equation 
p  =  Ri2,  the  average  rate  of  heat  production  must  be  proportional 
to  the  average  value  of  i2.  The  average  heating  effect  deter- 
mines the  deflection  of  such  an  instrument  as  a  hot-wire  ammeter, 
which  thus  indicates  a  current  /  fulfilling  the  condition,  average 
power  =  RI2.  The  indicated  current  /  must  therefore  be  the 
square  root  of  the  average  value  of  &.  The  square  root  of  the 
mean  square  of  an  alternating  current  or  emf  is  called  the  "effec- 
tive "  or  "  root-mean-square  "  current  or  emf.  All  ordinary  amme- 
ters and  voltmeters  used  in  alternating-current  measurements 
give  effective  values. 

When  an  electromotive  force  has  the  sine-wave  form,  e=E0 
sin  cot,  the  mean  square  value  is  proportional  to  the  average 
value  of  sin2  wt  during  a  half  cycle,  which  is  equal  to  0.5.  The 
effective  value  is  proportional  to  the  square  root  of  this,  so  that 
the  effective  value  is  V^5  E<»  or 

E  =  0.707  E0 
Similarly  in  the  case  of  current, 

7  =  0.707  70 

8.  CIRCUIT  HAVING  RESISTANCE  AND  INDUCTANCE 

When  an  emf  is  suddenly  impressed  on  a  circuit  containing 
inductance  as  well  as  resistance,  say  by  closing  a  switch,  a  cur- 
rent begins  to  flow  but  does  not  rise  to  its  full  value  instantly. 
The  magnetic  flux  accompanying  the  current  causes  a  self -induced 
emf  which  by  Lenz's  law  opposes  the  increase  of  current.  There 
are  then  acting  in  the  circuit  two  emf's,  the  impressed  emf  e  and 

di 
the  emf  of  self-induction,  which  by  equation  (8)  is  —L-rf       The 

minus  sign  is  used  to  indicate  that  the  induced  emf  opposes  the 
impressed  emf. 

This  is  similar  to  the  action  of  a  mechanical  force  on  a  mate- 
rial object;  the  applied  force  is  opposed  by  the  force  of  inertia 
and  some  time  is  required  before  the  body  moves  with  the  final 
velocity  determined  by  the  applied  force  and  the  friction.  The 
opposing  force  of  inertia  in  the  mechanical  case  is  given  by  the 


Circular  of  the  Bureau  of  Standards 


product  of  mass  by  the  time  rate  of  change  of  velocity.  The  force 
of  inertia  corresponds  to  induced  emf,  the  mass  to  inductance, 
and  velocity  to  current. 

The  total  emf  acting  to  produce  current  through  the  resistance 
is  the  sum  of  the  impressed  emf  and  the  emf  of  self-induction, 
thus: 

e-L%-Ri 

at 

This  equation  gives  the  relation  between  current  and  applied  emf 
at  any  instant.  It  is  usually  convenient  to  consider  this  equation 
in  the  form 


4 
at 


(12) 


which  indicates  that  the  applied  emf  is  opposed  by  the  resistance 
and  the  inductance.  When  R  is  relatively  large  or  L  relatively 
small  the  current  comes  very  quickly  to  its  final  value. 


FIG.  9. — Circuit  with  resistance  and  inductance  in  series 

Impedance. — In  alternating-current  and  radio  work  the  most 
common  and  the  simplest  type  of  electromotive  force  is  the  sine 
wave.  Such  an  emf  is  expressed  by  equation  (10).  Supposing  a 
sine-wave  emf  to  be  impressed  on  a  circuit,  equation  (12)  becomes 


di 

-r.  +  Ri  =  E0  sin  ut 


d3) 


The  solution  of  this  differential  equation  (neglecting  a  term  which 
represents  the  transient  phenomena  when  the  current  is  started)  is 


(R  sin  uit  —  coL  cos 


or 


+0)2L2 


sin  (ut  —  8) 


(14) 
d5) 


Radio  Instruments  and  Measurements  25 

where  6  is  defined  by 

tan0=~  (16) 

The  current  which  flows  in  a  circuit  containing  constant  resistance 
and  inductance  due  to  a  sine  emf  is  also  a  sine  wave.  The  emf 
varies  as  sin  (wf),  the  current  as  sin  (at  —  6),  hence  the  current 
lags  behind  the  emf  by  the  angle  6.  This  angle  is  called  the  phase 
angle.  The  instantaneous  current  becomes  a  maximum  (70)  when 
sin  (at  —  6)  —  i , 

E0 


and,  since  the  effective  values  /  and  E  are  equal  to  0.707  70  and 
0.707  Eo,  respectively, 


This  has  the  form  of  Ohm's  law,  the  quantity  ^2  +w2L2  occurring 
in  place  of  R.  This  quantity  is  for  this  circuit  the  value  of  the 
impedance,  which  is  defined  as  the  ratio  of  emf  to  current.  Since 
w  =  27r  times  the  frequency  it  is  clear  that  impedance  is  a  function 
of  frequency  as  well  as  of  resistance  and  inductance. 

Power.  —  The  power  expended  in  the  circuit  is  at  any  instant 
the  product  of  the  instantaneous  electromotive  force  and  current; 
p  =  ei.  The  average  power  is  the  mean  taken  over  a  complete 
cycle  of  this  instantaneous  product.  Performing  the  calculation, 
the  average  power  is  found  to  be 

P  =  E7cos0, 

where  E  and  7  are  effective  values  and  6  is  the  phase  angle,  defined 

p 
by  equation  (16)  above.     The  ratio      .  =  cos  6  is  called  the  power 


factor  of  the  circuit. 

9.  CIRCUIT  HAVING  RESISTANCE,  INDUCTANCE  AND  CAPACITY 

When  a  circuit  contains  a  condenser  in  series  with  resistance 
and  inductance  the  applied  electromotive  force  is  opposed  by  the 
potential  difference  of  the  condenser  in  addition  to  the  opposition 
of  the  resistance  and  inductance.  The  potential  difference  of  the 


condenser  equals       which  may  be  written  *j=     so  that  ecluation 
(12)  becomes 

di    fidt  ,     . 


26  Circular  of  the  Bureau  of  Standards 

Taking  e  =  E0  sin  cot,  and  differentiating  the  equation, 
dH        di     i 


The  solution  (neglecting  terms  representing  the  transients,  which 
die  out  very  quickly  after  the  current  is  started)  is 


E0 


sin  (ut  —  8) 


The  phase  angle  6  is  given  by 


a 
tan  0  =  -Fr- 


R      RaC 
R 


d9) 


FIG.  10. — Circuit  with  resistance,  inductance,  and  capacity  in  series;  a  typical  radio  circuit 

The  maximum  value  of  the  current  is 

E0 


/0  = 


The  impedance  is 


V- 


The  relation  between  effective  current  and  emf  is 

E 


(20) 


Radio  Instruments  and  Measurements 


27 


It  is  to  be  noted  that  the  terms  in  L  and  C  have  opposite  signs 
in  this  equation.  Thus  one  tends  to  neutralize  the  other,  and 
comparing  with  equation  (17)  for  a  circuit  with  resistance  and 
inductance  only,  the  impedance  of  an  inductive  circuit  can  be 
reduced  and  the  current  increased  by  putting  a  condenser  of 
suitable  value  in  series  with  the  inductance.  This  increase  of  the 
current  has  sometimes  been  called  resonance  or  partial  resonance, 
but  the  term  "  resonance"  is  usually  reserved  for  the  production  of 
a  maximum  current,  as  treated  in  section  1 1 ,  below. 

Special  Cases. — It  is  of  interest  to  consider  the  following  special 

cases : 3 

I.  L  =  o  and  C  =  oo . 
II.  C=co. 
III.  L  =  o. 

Case  I  represents  a  circuit  with  resistance  alone.     The  equations 

£ 

just  above  give  for  this  case  I  =  -~,  8  =  0°.  The  impressed  electro- 
motive force  and  the  current  are  in  phase,  and  their  ratio  is  the 
resistance,  just  as  with  direct  current. 


FIG.  n.i — Circuit  with  resistance  and  capacity  in  series 

Case  II  is  that  of  a  circuit  with  resistance  and  inductance, 
which  has  already  been  treated  in  section  8. 

Case  III  is  that  of  a  circuit  with  resistance  and  capacity  in 
series.  Equations  (20)  and  (19)  give 

E 


tan  B  =  - 


RaC 


8  Putting  C=oo  is  mathematically  equivalent  to  the  statement  that  the  condenser  is  short-circuited. 
As  the  distance  between  the  plates  of  a  condenser  is  decreased  the  capacity  increases  without  limit.  We 
may  consider,  then,  that  when  the  plates  touch  together  and  the  condenser  is  short-circuited  the  capacity 
is  infinite. 


28 


Circular  of  the  Bureau  of  Standards 


The  case  is  of  special  importance  when  the  resistance  term  is 
entirely  due  to  energy  losses  within  the  condenser.  It  is 
frequently  convenient  to  deal  with  the  "phase  difference"  ^, 
which  =  90°  —  6,  rather  than  with  the  phase  angle.  The  phase 
difference  is  given  by 

tan  \fr  =  — 


If  R  is  very  small  tan  \f/  =  ^,  and  the  phase  difference  =  —  RwC. 

10.  "VECTOR"  DIAGRAMS 

Writing  equation  (17)  in  the  form  E  =  ^/R2P  +w2L2P,  it  is  evi- 
dent that  E  has  such  a  value  as  would  be  given  by  the  diagonal  of 
a  rectangle  having  sides  equal  to  RI  and  coL/.  It  is  therefore  pos- 
sible to  determine  the  value  of  E  by  the  aid  of  a  vector  diagram 


RI 


cuLI 


c»LI 


FIG.  12. — Vector  combination 
of  electromotive  forces 


FIG.  13. — Vector  diagram  for  resistance  and  in- 
ductance in  series 


such  as  is  used  for  calculating  the  resultant  of  mechanical  forces. 
u>LI  is  represented  as  a  vector  perpendicular  to  RI,  and  their  re- 
sultant is  E.  The  current  /  is  represented  as  a  vector  in  the  same 
direction  as  RI;  since,  from  equations  (10),  (15),  and  (16),  the 
current  and  electromotive  force  differ  in  phase  by  the  angle 

whose  tangent  is  -=-,  and  this  is  equal  to  the  angle  6  in  Fig.  13. 
If  equation  (20)  is  written  in  the  form 


E  = 


it  is  evident  that  E  is  calculable  as  the  result  of  adding  the  three 
vectors  RI,  wLI,  and  — ^,  drawn  as  in  Fig.  14. 


Radio  Instruments  and  Measurements 


29 


These  three  quantities  are  the  emf 's  across  the  resistance,  induc- 
tance, and  capacity,  respectively.  The  emf  — ^  is  drawn  downward 

from  the  origin,  opposite  in  direction  to  uLI,  corresponding  to  the 
minus  sign  in  the  equation.  The  phase  angle  between  /  and  the 
resultant  E  is  6.  From  the  Fig.  14, 


tan0  = 


I 

^c 


(21) 


R 


in  agreement  with  (19)  above.     When  coL  is  greater  than  —^,  E 

OIL, 

is  above  the  horizontal  line  as  shown,  6  is  positive,  and  the  cur- 


coLI 


J_ 

cuC 


e\ 


RI 


FIG.  14. — Vector  diagram  for  resistance,  inductance,  and  capacity  in  series 


rent  is  said  to  lag  behind  the  electromotive  force.     When  —~  is 

coC 

greater  than  coL,  8  is  negative,  and  the  current  is  said  to  lead  the 
electromotive  force.  The  component  of  emf  in  phase  with  the 

current  is  RI.  The  component  at  90°  to  the  current  is  (  uL  — ^  j  7. 
The  ratio  of  this  component  emf  to  the  current  is  called  the 
reactance.  Its  value  here  is  (coL — -^j.  The  ratio  of  the  re- 
sultant emf  to  the  current  is  called  the  impedance,  which  is  here 
equal  to  J R2  -fv£«L-^V 

If  all  the  electromotive  forces  in  Fig.  14  be  divided  by  /,  the 
component  vectors  then  become  resistance  and  reactances  and 


3O  Circular  of  the  Bureau  of  Standards 

these  combine  vectorially  to  give  the  impedance  as  a  resultant/ 
It  is  sometimes  convenient  to  speak  of  resistance  and  reactance 
as  impedance  components.  Reactance  (usually  denoted  by  the 
symbol  X)  is  expressible  in  ohms  just  as  resistance  is.  The 
reactance  in  the  case  under  consideration  consists  of  two  parts, 
the  "inductive  reactance"  coL  and  the  "capacitive  reactance" 

—~.     These  may  be  denoted,  respectively,  by  XL  and  Xc.     From 

the  expression  (21)  above  it  is  seen  that  the  tangent  of  the  phase 
angle  is  a  ratio  of  impedance  components. 

X        X^L  —  Xc 

SS~R  =     -  R 

When  the  capacitive  reactance  is  greater  than  the  inductive 
reactance,  the  total  reactance  X  and  the  phase  angle  have  negative 

R 


HWiiH 


FIG.  15. — Circuit  with  resistance  and  capacity  in  parallel 


values.     In  any  case  the  ratio  of  reactance  to  resistance  is  the 
tangent  of  the  phase  angle. 

Phase  Difference. — In  a  circuit  consisting  of  resistance  and  capa- 
city only,  or  resistance  and  inductance  only,  in  series,  it  is  more 
convenient  to  deal  with  the  phase  difference  than  with  the  phase 
angle.  The  tangent  of  the  phase  difference  is  the  reciprocal  of  the 
tangent  of  the  phase  angle.  When  an  angle  is  small  it  is  equal  to 
its  tangent,  and  consequently  the  phase  difference  is  equal  to  the 
ratio  of  resistance  to  reactance,  in  a  series  circuit  in  which  the  re- 
sistance is  small  compared  with  the  reactance.  This  is  in  agree- 
ment with  the  case  discussed  above  on  page  27,  where  it  was  shown 
that  the  phase  difference  of  a  condenser  with  resistance  in  series 


Radio  Instruments  and  Measurements  3 1 

=  —RcoC.     Similarly,  the  phase  difference  of  an  inductance  with 

resistance  in  series  =  — r* 
coL 

Vector  Addition  of  Currents. — For  series  circuits  the  emfs  and 
the  components  of  impedance  combine  vectorially  just  as,  with 
direct  current,  the  emfs  and  resistances  combine  algebraically. 
With  direct  current  in  parallel  circuits,  on  the  other  hand,  the 
currents  and  conductances  add  up  algebraically.  For  parallel 
circuits  with  alternating  current,  the  currents  combine  vectorially, 
and  so  do  the  components  of  the  ad- 
mittance (reciprocal  of  impedance) . 

Suppose,  for  example,  a  condenser  and 
resistance  in  parallel  (Fig.  1 5) .  Current  /  is 

£ 

the  vector  sum  of  the  currents  7R  =  ~  an(i 
7C  =  wCE.    The  impressed  emf  and  the  _ 

FiG.  io. —  v ector diagram  for  re- 

Current  =    are     in     the     Same     direction,    Distance  and  capacity  in  parallel 
K. 

while  the  current  uCE  leads  the  impressed  emf  by  90°.     The 
resultant  current  /  leads  E  by  the  angle  6,  where  tan  6  = 
From  Fig.  16, 


and  the  admittance  (ratio  of  resultant  current  to  emf)  is  \  -&-2  +  co2C3. 


11.  RESONANCE 


In  a  circuit  consisting  of  inductance,  capacity  and  resistance 
in  series,  the  effective  current  has  been  shown  to  be 


L__LY  (22> 

When  coL  =  — ^»  the  impedance  is  a  minimum  and  the  effective 
coC 

current  is  a  maximum.  This  condition  for  maximum  current  is 
called  resonance.  The  ratio  of  the  current  at  resonance  to  the 
current  in  the  circuit  with  the  condenser  removed  has  been  called 
the  "resonance  ratio";  this  quantity  is  practically  the  same  in 
radio  circuits  as  the  "sharpness  of  resonance"  denned  below. 

At  a  given  "frequency  resonance  may  be  brought  about  by 
varying  either  the  capacity  or  the  inductance.     On  the  other 

35601°— 18 3 


32  Circular  of  the  Bureau  of  Standards 

hand,  for  a  circuit  of  given  L  and  C,  there  is  some  particular 
frequency  at  which  resonance  occurs.     The  condition 


is  equivalent  to 


i 
^C 


LC 

i 


(23) 


(24) 


The  relation  (24)  is  of  the  greatest  importance  in  high-frequency 
work.  It  is  the  fundamental  equation  of  the  wave  meter,  for 
instance.  A  number  of  other  important  ways  of  expressing  the 
same  relation  are  given  in  section  78. 

Simplified  Current  Equation  at  Resonance.  —  At  resonance  the 
inductive  reactance  is  equal  to  the  capacitive  reactance,  the  total 
reactance  is  zero,  and  the  impedance  equals  simply  the  resistance. 
That  is,  at  resonance,  equation  (22)  reduces  to 


- 
r~R 

This  means  that  the  impressed  emf  is  strictly  equal  to  RIT.  The 
potential  difference  across  the  condenser  and  that  across  the 
inductance  may  be  greater  than  this,  and  in  fact  may  be  many 
times  the  impressed  emf.  Being  equal  and  opposite,  they 
neutralize  each  other  and  contribute  nothing  to  the  resultant 
emf  opposing  the  applied  emf. 

Mechanical  Illustration.  —  The  phenomenon  of  resonance  is  well 


FIG.  17. — Simple  mechanical  system  which  can 
exhibit  the  phenomenon  oj  resonance 

illustrated  by  the  vibration  of  a  spring  with  a  mass  attached. 
When  a  force  F  acts  on  the  mass  m,  it  is  opposed  by  the  stiffness 
of  the  spring,  by  the  inertia  of  the  mass,  and  by  friction.  The 
analogy  to  the  electrical  case  is  not  perfect,  since  friction  due  to 
sliding  is  not  proportional  to  the  velocity.  If  the  force  is  applied 
periodically,  there  will  be  a  certain  particular  frequency  for 


Radio  Instruments  and  Measurements  33 

which  a  more  vigorous  oscillation  is  produced  than  for  any  other. 
When  the  frequency  of  the  applied  force  is  just  equal  to  the 
frequency  of  resonance,  the  applied  force  is  all  used  in  overcoming 
friction;  the  elasticity  of  the  spring  and  the  inertia  of  the  mass 
constitute  two  equal  forces  opposing  each  other.  These  two 
opposite  forces  may  be  much  greater  than  the  applied  force. 
For  instance,  the  vibration  may  become  so  violent  as  to  break 
the  spring  although  the  impressed  force  is  far  too  small  to  do  so. 

Magnification  of  Voltage. — Similarly,  in  the  case  of  the  electrical 
circuit,  there  is  danger  of  breaking  down  the  condenser  in  a 
resonant  circuit  because  the  potential  difference  across  the  con- 
denser is  much  greater  than  the  applied  electromotive  force. 
The  ratio  of  the  voltage  across  the  condenser  to  the  applied 
voltage  is  greater  the  smaller  the  resistance  (including  under 
resistance  not  only  the  ordinary  "ohmic"  resistance  of  the  con- 
ductors but  all  sources  of  energy  loss,  such  as  dielectric  loss  in 
the  condenser).  In  comparing  the  electrical  circuit  with  the 
vibrating  spring,  one  should  remember  that  the  mass  is  the  analog 
of  the  inductance  and  the  spring  the  analog  of  the  condenser.  It 
is  unfortunate  that  the  diagram  generally  used  for  an  inductance 
is  the  same  as  that  used  above  for  a  spring. 

Vector  Diagram  of  Resonance. — Resonance  phenomena  are 
shown  in  an  interesting  manner  by  means  of  vector  diagrams. 
In  Fig.  14,  which  illustrates  the  vector  diagram  of  emfs  for  a 
circuit  with  resistance,  inductance,  and  capacity  in  series,  the 
inductive  reactance  coL  is  taken  to  be  greater  numerically  than 

the  capacitive  reactance  -^  so  that  the  resultant  vector  E  has  a 

direction  corresponding  to  a  positive  rotation  from  the  direction 
of  RI  through  an  angle  6.  Suppose  that  the  frequency  is  de- 
creased; ooL  decreases  and  — ^  increases  numerically.  When 

they  become  equal,  the  total  reactance  ( coL -~  j  and  the  angle 

6  become  zero,  and  E  equals  RI.  The  diagram  then  becomes 
Fig.  i 8. 

Resonance  Curves. — As  already  stated,  in  a  circuit  of  given  L 
and  C  there  is  some  frequency  at  which  resonance  occurs,  given 

bycoL=— -~-  At  all  other  frequencies  the  inductive  reactance 
and  the  capacitive  reactance  are  unequal,  and  their  difference 


34 


Circular  of  the  Bureau  of  Standards 


enters  the  expression  for  impedance.  At  frequencies  less  than 
the  frequency  of  resonance  the  capacitive  reactance  is  the  larger, 
and  consequently  we  may  say  that  at  low  frequencies  the  capacity 
keeps  down  the  current,  while  at  high  frequencies  it  is  the  induc- 
tance that  keeps  the  value  of  the  current  down.  For  any  depar- 
ture from  the  condition  of  resonance,  whether  by  variation  of  fre- 
quency, of  inductance,  or  of  capacity,  the  current  is  diminished. 
The  process  of  varying  either  the  capacity  or  the  inductance  to 
obtain  the  setting  at  which  the  circuit  is  in  resonance  with  the 
frequency  of  the  applied  electromotive  force  is  called  "tuning" 
the  circuit  or  tuning  the  circuit  to  resonance. 

The  reduction  of  the  current  on  both  sides  of  resonance  is 
shown  in  Fig.  19,  in  which  the  square  of  current  is  plotted  against 
capacity,  the  emf  being  constant.  Such  curves  are  called  reso- 
nance curves.  • 


o>LI 


RI 


cuC 


FIG.  18. — Vector  diagram  of  series  circuit  in  resonance 

The  three  curves  shown  are  for  an  actual  circuit,  with  its  normal 
resistance  of  4.4  ohms,  with  5  ohms  added,  and  with  10  ohms 
added.  The  inductance  is  377  microhenries  and  the  frequency 
169  100  cycles  per  second.  The  curves  show  theoretical  values 
as  obtained  from  the  formula 

E2 


The  theoretical  values  were  closely  checked  by  actual  observa- 
tions, using  a  pliotron  as  a  source  of  alternating  current  and  a 
thermocouple  and  galvanometer  to  measure  the  current.  The 


Radio  Instruments  and  Measurements 


35 


square  of  current  is  plotted  instead  of  current,  simply  because 
the  galvanometer  deflections  were  proportional  to  the  square  of 
current.  The  ordinates  are  thus  in  terms  of  galvanometer  deflec- 
tions. 

The  value  of  the  constant  impressed  emf  is  given  by  E  =  RIT, 
where  7r  =  current  at  resonance.  In  the  arbitrary  units  resulting 
from  the  expression  of  current-square  in  galvanometer  deflections 
and  R  in  ohms,  for  curve  A  the  value  of  E  =  ^X  1/19  =  19.2. 
The  emf  across  the  inductance  =  coL7,  so  that  at  resonance  its 


FIG.   19.  —  Resonance  curves  for  series  circuit 
•with  different  resistances 


value  is  2ir  x  169  100  x  377  X  io~6  X  V^9  —  J  75°-  The  emf  across 
the  condenser  at  resonance  =  —~  ,  which  equals  the  same  value, 

1750.  Note  that  this  is  much  greater  than  the  applied  emf,  19.2. 
In  the  case  of  curve  B,  the  applied  emf  is  the  same,  but  the  emf 
across  the  inductance  and  the  equal  emf  across  the  condenser  at 
resonance  =  27r  Xi  69  100X377  X  io~6X  V4-i6  =  8i8.  ln  the  case 
of  curve  C,  the  equal  emfs  across  inductance  and  capacity  at 
resonance  each  equal  534.  The  applied  emf  having  the  same 
value,  19.2,  for  each  curve,  this  clearly  illustrates  the  statement 
previously  made  that  the  ratio  of  condenser  voltage  to  applied 
voltage  is  greater  the  smaller  the  resistance. 


36  Circular  of  the  Bureau  of  Standards 

Sharpness  of  Resonance. — One  of  the  principal  applications  of 
the  phenomenon  of  resonance  is  the  determination  of  frequency. 

Since  the  current  is  a  maximum  when  w  =    , — •,  the  frequency 

VLC 

is  determined  when  L  and  C  are  known.  The  precision  with  which 
frequency  can  be  determined  by  this  method  depends  upon  the 
sensitiveness  of  the  current  indication  to  a  given  change  in  C  or 
L  at  resonance.  This  sensitiveness  is  obviously  greater  the 
sharper  the  peak  (Fig.  19).  The  precision  of  determination  of 
frequency,  therefore,  depends  upon  what  may  be  called  the 
sharpness  of  resonance,  a  quantity  which  measures  the  fractional 
change  in  current  for  a  given  fractional  change  in  either  C  or  L 
at  resonance.  (This  quantity  has  also  been  called  selectivity; 
see  also  statement  on  p.  31  regarding  the  term  resonance  ratio.) 
The  sharpness  of  resonance  is  an  important  characteristic  of  a 
circuit  and  is  very  simply  related  to  the  phase  differences  and 
other  constants.  It  may  be  denned  in  mathematical  terms  by 
the  following  ratio 


±(C,-Q 
C 

where  the  subscript  r  denotes  value  at  resonance,  and  7t  is  some 
value  of  current  corresponding  to  a  capacity  C  which  differs  from 
the  resonance  value.  The  numerator  of  this  expression  is  some- 
what arbitrarily  taken  to  be  the  square  root  of  the  fractional 
change  in  the  current-square  instead  of  taking  directly  the 
fractional  change  of  the  first  power  of  current.  This  is  done 
because  of  the  convenience  in  actual  use  of  this  expression  (since 
the  deflections  of  the  usual  detecting  devices  are  proportional  to 
the  square  of  the  current)  and  also  because  of  its  mathematical 
convenience.  It  is  readily  shown  that  the  sharpness  of  resonance 
thus  defined  is  equal  to  the  ratio  of  the  inductive  reactance  to 
the  resistance.  Since 

«/.-£  (25) 

the  relation 


r  2  _ . 


Radio  Instruments  and  Measurements  37 

becomes 

772 


,o>Cr 
This,  together  with 

gives 


(26) 


The  right-hand  member  of  the  equation  is  the  ratio  of  the  capaci- 
tive  reactance  at  resonance  to  the  resistance.  In  virtue  of  equa- 
tion (25),  it  is  also  equal  to  the  ratio  of  the  inductive  reactance 

to  the  resistance;  thus  sharpness  of  resonance  =  -~-' 

It  is  of  interest  to  note  the  relation  of  the  sharpness  of  reso- 
nance to  phase  difference.  As  shown  above  on  page  31,  the  phase 
difference  of  a  series  combination  either  of  resistance  and  induct- 
ance or  of  resistance  and  capacity  is  equal  to  the  ratio  of  resist- 
ance to  reactance.  If  in  a  circuit  having  an  inductance  coil  and 
a  condenser  in  series  the  only  resistance  is  that  of  the  inductance 
coil,  it  follows  that  the  sharpness  of  resonance  is  equal  to  the 
reciprocal  of  the  phase  difference  of  the  coil.  If,  on  the  other  hand, 
the  resistance  of  the  circuit  is  all  due  to  energy  loss  in  the  con- 
denser, the  sharpness  of  resonance  is  equal  to  the  reciprocal  of 
the  phase  difference  of  the  condenser.  A  measurement  of  the 
sharpness  of  resonance  thus  gives  the  phase  difference  directly. 
If  the  resistance  is  partly  in  the  coil  and  partly  in  the  condenser, 
each  has  a  phase  difference  and  the  sharpness  of  resonance  is 
equal  to  the  reciprocal  of  the  sum  of  the  two  phase  differences. 


Circular  of  the  Bureau  of  Standards 


It  has  been  mentioned  that  there  is  danger  of  breaking  down  a 
condenser  in  a  resonant  circuit  because  the  potential  difference 
across  the  condenser  may  be  many  times  higher  than  the  applied 
electromotive  force.  This  danger  is  directly  in  proportion  to  the 
sharpness  of  resonance,  or  inversely  as  the  phase  difference  of  the 
condenser.  For  if  the  applied  emf  is  RI,  the  condenser  voltage 

The  ratio  of  the  condenser  voltage  to  the 


at  resonance  is  — 


coCr 
i 


applied  emf  is  —  ^   ~  » which  is  the  sharpness  of  resonance  or  the 

reciprocal  of  the  condenser  phase  difference. 

Application  to  Radio  Resistance  Measurement. — Formula   (26) 
above,  which  gives  the  relation  between  the  sharpness  of  resonance 


FlG.  20. — Simple  circuit  for  measurements  of  resist- 
ance or  wave  length 

and  the  phase  difference  of  the  condenser,  has  been  shown  to  have 
important  applications  to  the  precision  of  frequency  measure- 
ment and  to  the  rise  of  voltage  on  the  condenser.  Another  appli- 
cation of  great  importance  is  the  measurement  of  resistance. 
This  is  seen  by  writing  the  equation  in  the  form 

±(c,-o  [-77-  (27) 

Thus  the  resistance  of  a  simple  circuit  as  in  Fig.  20  is  measured 
by  observing  deflections  of  the  indicating  instrument  A  for  two 
settings  of  the  variable  condenser,  one  setting  at  resonance  CT 
and  any  other  setting  C.  This  is  one  of  the  principal  methods 
of  measuring  high-frequency  resistance,  and  may  be  called  the 
' '  reactance  variation ' '  method.  Other  ways  of  using  the  prin- 
ciple of  reactance  variation  are  described  in  section  50  below. 


Radio  Instruments  and  Measurements 


39 


The  method  is  rigorous,  involving  no  approximations,  provided 
the  applied  emf  is  undamped.  The  resistance  so  measured  is  the 
effective  resistance  of  the  entire  circuit,  including  that  due  to 
condenser  losses  and  radiation. 

12.  PARALLEL  RESONANCE 

When  a  coil  and  a  condenser  are  in  parallel  in  a  circuit,  the 
phenomena  are  strikingly  dif- 
ferent from  those  of  the  series 
arrangement.  The  total  cur- 
rent /  is  the  vector  sum  of  the 
currents  in  the  two  branches, 
7L  and  7C.  The  current 
through  the  coil  depends  on- 
its  resistance  and  inductance, 
thus, 

E 


Also 


FIG.  21. — Parallel  circuit  having  capacity  in 
parallel  with  inductance  and  resistance;  un- 
der certain  conditions  the  current  in  either 
of  the  branches  exceeds  that  in  the  main  line 


assuming  the  condenser  loss  to  be  negligible.     Taking  the  sum  of 

the  two  currents,  with  due  regard  to  their 
•^  phase  relation,  the  total  current  is 


/-£ 

When 


//  coL     V     /       R       V 

V  \      ~R2  +  tfU)     \R*  4*VLV 


(28) 


the  total  current  is  in  phase  with  the  emf, 
and  has  the  value, 


ER 


(29) 


This  is  the  minimum  current  for  varying 
values  of  C  and  is  very  nearly  the  minimum 
current  for  varying  values  of  L  or  w. 

Equation  (28)  is  the  condition  for  what 
may  be  called  inverse  resonance  or  parallel 
resonance.  At  parallel  resonance,  the 
total  current  in  the  external  circuit  is  less  than  the  current  in  the 
coil.  This  is  because  the  currents  in  condenser  and  coil  are  in 


FIG.  22. — Vector  diagram 
for  parallel  circuit 


40  Circular  of  the  Bureau  of  Standards 

opposite  directions  as  regards  the  external  circuit,  and  thus  tend 
to  neutralize  each  other  in  that  circuit. 

Simple  Case. — When  the  resistance  of  the  coil  is  very  small 
compared  with  its  reactance,  as  is  usual  at  radio  frequencies, 
equation  (28),  the  condition  for  parallel  resonance,  becomes 


(using  co0  to  denote  the  value  of  o>  at  parallel  resonance) ,  or 

i 

Wo  ==      /  _    _ 

VLC 

The  total  current  at  this  frequency  is 

ER 


(30) 


FIG.  23. — Vector  diagram  illustrating 
resonance  in  simple  parallel  circuit 


o  i  ex  /owia 

FIG.  24. — Resonance  curve  showing  the  con- 
dition of  parallel  resonance 


The  current  in  the  condenser  is  very  closely  equal  to  that  in  the 

£ 

coil,  the  value  being  — T'     The  total  current  is  the  vector  sum 

C00i^ 

of  the  currents  in  coil  and  condenser,  and  is  thus  smaller  than  the 

r> 

current  in  either  by  the  ratio  — j-     As  suggested  by  a  compari- 

COo/-* 

son  of  Fig.  22  and  Fig.  23  the  resultant  current  may  be  yanish- 
ingly  small.     The  combination  of  coil  and  condenser  acts  like  a 


Radio  Instruments  and  Measurements  41 

very  large  impedance  in  the  main  circuit,  the  value  of  this  im- 
pedance being  — ^ — .  For  any  variation  of  frequency,  induc- 
tance, or  capacity,  from  the  condition  of  parallel  resonance,  the 
total  current  increases. 

At  low  frequencies  the  inductance  carries  the  greater  part  of  the 
current,  and  at  high  frequencies  the  condenser  is  the  more  im- 
portant. 

Comparison  of  Series  and  Parallel  Resonance. — It  is  interesting 
to  compare  the  resonance  phenomena  in  a  series  circuit  with  the 
phenomena  of  parallel  resonance.  In  the  former  case  the  indi- 
vidual voltages  across  the  coil  and  condenser  exceed  the  result- 
ant voltage  across  both,  whereas  in  the  latter  case  the  separate 
currents  exceed  the  resultant  current.  The  impedance  intro- 
duced into  the  circuit  by  the  series  combination  is  yanishingly 

small,  and  the  impedance  due  to  the  parallel  combination  is  very 

17 
large.     Comparing  equation  (29)  with/=    ,  .»  the  ratio' of 

the  total  current  at  parallel  resonance  to  the  current  in  the  circuit 

r> 

with  the  condenser  removed  is    /r.,          =•    Thus   the  current   is 


reduced  in  parallel  resonance  in  the  same  ratio  that  it  is  increased 
in  series  resonance.  The  two  kinds  of  resonance  are  discussed 
further  in  the  next  section. 

RADIO    CIRCUITS 
13.  SIMPLE  CIRCUITS 

A  typical  radio  circuit  comprises  an  inductance  coil,  a  con- 
denser, and  a  source  of  electromotive  force,  in  series. 

The  source  E  may  be  a  small  coil  in  which  an  alternating  electro- 
motive force  is  induced  by  the  current  in  a  neighboring  circuit. 
Some  of  the  phenomena  in  such  a  circuit  have  already  been  treated 
under  "The  principles  of  alternating  currents."  What  is  there 
given  applies  to  high  as  well  as  low  frequencies.  Some  of  the 
phenomena  and  their  mathematical  treatment  are  much  simplified 
at  high  frequencies.  Electromotive  forces  of  sine-wave  form  are 
assumed  in  this  discussion;  the  results  obtained  apply  equally  to 
slightly  damped  waves. 

It  will  be  recalled  that  the  reactance  of  an  inductance  is  uL, 

and  the  reactance  of  a  capacity  is  — «•     It  is  essential  to  remember 

coC 


42  Circular  of  the  Bureau  of  Standards 

that  co  is  27T  times  the  frequency.  In  fact,  the  physical  meaning 
of  co  in  reactance  expressions  is  the  same  thing  as  frequency;  the 
27T  is  a  factor  result  ing  from  the  way  the  units  are  defined.  The 
expression  coL  tells  us  that  the  reactance  of  an  inductance  is  pro- 
portional to  frequency. 

Series  Circuit. — The  simple  circuit  of  Fig.  25  is,  in  fact,  the 
principal  circuit  used  in  radio  transmitting  sets,  receiving  sets, 
and  wave  meters.  Some  of  its  chief  properties  are  conveniently 
brought  out  by  a  graphical  study  of  the  variation  of  its  reactance 
with  frequency.  Advantage  is  taken  of  the  fact  that  resistance 
is  a  negligible  part  of  the  impedance  (except  at  resonance),  to 
obtain  very  simply  an  idea  of  the  way  the  current  varies  with 
frequency.  Small  current  corresponds  to  large  reactance  (either 
positive  or  negative),  and  vice  versa. 


FIG.  25. — Simple  series  circuit 


The  reactance  of  the  circuit  is  ( coL  — -?=,)•  The  inductive  react- 
ance, coL,  is  the  predominating  portion  of  this  at  high  frequencies. 
It  is  represented  by  the  line  coL  in  Fig.  26.  The  capacitive  react- 
ance predominates  at  low  frequencies;  it  is  represented  by  the 

line  —~'    The  sum  of  these  two  is  represented  by  the  line  marked 


total  reactance, 
crosses  the  axis- 


At  the   point   co' 


-i.  e.,  where  coL  = 


coC 


where  this  reactance  curve 
— the  current  is  a  maximum. 


The  resonance  curve,  showing  variation  of  the  current  with  fre- 
quency, would  rise  to  infinity  at  the  point  co',  where  the  total 
reactance  is  o,  if  it  were  not  for  the  resistance  in  the  circuit. 
While  the  current  at  resonance  is  determined  by  the  resistance, 
the  frequency  of  resonance  is  given  accurately  by  the  reactance 
curve  which  takes  no  account  of  resistance.  The  most  important 


Radio  Instruments  and  Measurements 


43 


aspect  of  resonance  phenomena  is  thus  shown  by  the  simple  react- 
ance curve,  the  plotting  of  the  current  curve  being  unnecessary. 
Use  of  Reactance  Curves. — Complex  circuits  can  be  studied  and 
much  useful  information  easily  obtained  by  the  use  of  reactance 
curves.  The  effect  of  any  auxiliary  circuit  upon  a  wave  meter  or 
a  transmitting  apparatus  can  be  determined,  as  will  be  shown 
later.  In  any  such  diagram  the  points  where  the  reactance  curve 
crosses  the  co  axis  give  the  frequencies  at  which  the  current  is  a 
maximum. 


2000 


600 


500 


2000 


1500 


FIG.  26. — Reactance  diagram  for  simple  series  circuit,  showing  the  capacitive  and  induc- 
tive reactances  and  their  resultant  at  different  frequencies .  The  current  in  the  circuit 
is  a  maximum  at  uf 

Parallel  Circuit. — An  inductance  and  a  capacity  placed  in  paral- 
lel in  a  circuit  behave  very  differently  from  the  series  arrangement 
of  inductance  and  capacity  already  discussed.  As  shown  in  Fig. 
27,  the  same  electromotive  force  is  impressed  upon  the  terminals 
of  both  L  and  C  by  the  source  E,  which  may  be  a  spark  gap, 
another  condenser,  a  coupling  coil,  etc.  The  total  current  /  is  the 
sum  of  the  currents  in  L  and  C,  or 


7  =  4~-coCE 
coL 


/ 


The  ratio,  ^»  is  equal  to  the  reciprocal  of  the  impedance,  and  when 


44 


Circular  of  the  Bureau  of  Standards 


resistance  is  negligible  this  is  a  quantity  called  the  susceptance. 
The  total  susceptance  is  here  made  up  of  two  parts,  the  inductive 

susceptance  -y»  which  predominates  at  low  frequencies,  and  the 


FIG.  27. — Simple  parallel  circuit 

capacitive  susceptance  wC,  which  predominates  at  high  frequencies. 
Each  of  these  two  susceptances  is  the  reciprocal  of  the  correspond- 
ing reactance,  but  this  is  true  only  when  resistances  are  negligible. 


0.008 


FIG.  28. — Reactance  diagram  for  simple  parallel  circuit,  showing  the  capaciti-ve  and 
inductive  susceptances  at  different  frequencies,  together  -with  their  resultant  and  the 
resultant  reactance 

The  curve  marked  "Total  susceptance"  in  Fig.  28  was  obtained  by 

addition  of  the  two  curves,  —7-  and  coC.     The  curve  "Reactance " 

coL 

was  obtained  by  taking  reciprocals  of  the  points  on  the  curve  of  total 


Radio  Instruments  and  Measurements 


45 


susceptance.  The  reactance  of  the  circuit  is  small  at  very  low  and 
very  high  frequencies,  but  at  co0,  the  point  of  parallel  resonance, 
both  branches  of  the  reactance  curve  go  to  infinity.  The  current 
in  the  circuit  is  a  minimum  at  co0  and  would  be  strictly  zero  if 
there  were  actually  no  resistance  in  either  branch  of  the  circuit. 
Thus  it  is  seen  that  while  a  series  combination  of  inductance 

and  capacity  has  zero  reactance  when  wL  =  — ^  a  parallel  arrange- 
ment has  infinite  reactance  under  the  same  condition.  A  series 
arrangement  is  therefore  used  when  it  is  desired  to  make  current 
of  a  given  frequency  a  maximum,  and  a  parallel  arrangement  is 
used  when  it  is  desired  to  suppress  the  current  of  that  frequency. 

14.  COUPLED  CIRCUITS 

Circuits  which  are  more  complex  than  those  already  discussed 
may  be  considered  as  combinations  of  simple  circuits.  The 
component  simple  circuits  in  general  have  certain  parts  in  com- 


FIG.  29. — Simple  case  of  coupled  circuits  in  which  a 
parallel  circuit  u  combined  with  a  series  circuit 

mon,  and  these  parts  are  said  to  constitute  the  coupling  between 
the  circuits.  Suppose,  for  example,  the  simple  series  circuit  and 
the  simple  parallel  circuit  are  combined  as  shown  in  Fig.  29. 
The  coil  L  is  the  coupling  between  the  circuit  QL  and  the  circuit 
C2L. 

Elimination  of  Interference. — A  great  deal  of  information  about 
coupled  circuits  may  be  obtained  from  their  reactance  diagrams. 
A  curve  of  the  variation  of  reactance  with  frequency  tells  in  a 
very  simple  way  at  what  frequencies  the  current  is  either  large 
or  small.  The  reactance  of  C2  and  L  in  parallel  (Fig.  29)  is  as 
shown  in  Fig.  28  and  designated  by  X"  in  Fig.  30.  This  com- 
bination is  in  series  with  -C:.  Adding  the  curve  of  condenser 


46 


Circular  of  the  Bureau  of  Standards 


reactance— —  to  the  curve  X",  the  curve  X  is  obtained,  giving 
coCt 

the  reactance  to  current  flowing  through  the  ammeter.  At  the 
frequency  corresponding  to  «',  the  reactance  is  zero  and  the 
current  a  maximum.  At  co0  the  reactance  is  infinite  and  the 
current  is  a  minimum.  It  is  easily  seen,  therefore,  that  such  a 
circuit  is  very  useful  where  it  is  desired  to  have  current  of  a 
certain  frequency  in  a  circuit  but  to  exclude  current  of  a  certain 
other  frequency.  For  example,  if  it  is  desired  to  receive  radio 
messages  of  a  certain  wave  length  from  a  distant  station,  and  a 


1000 


750 


5OO 


250 


FIG.  30. — Reactance  diagram  for  combination  circuit  of  Fig.  29;  curve  X  is  the  resultant 

reactance  of  the  system, 

near-by  station  operating  on  a  different  wave  length  emits  waves 
so  powerful  as  to  interfere  with  the  reception,  the  interfering 
signals  can  be  greatly  reduced  by  using  this  kind  of  circuit.  The 
circuit  C2L  is  first  independently  tuned  to  resonance  with  the 
waves  which  it  is  desired  to  suppress.  The  setting  of  condenser 
Ci  is  then  varied  until  the  main  circuit  is  in  resonance  with  the 
desired  waves.  If  the  resistances  in  the  circuit  are  very  small, 
interference  is  readily  eliminated  in  this  manner.  The  same  thing 
is  accomplished  by  other  types  of  coupled  circuits,  as  explained 
below. 

Suppression  of  Harmonics. — Such   a  circuit  is  useful  also  in 
sending  stations  or  in  laboratory  set-ups,   where  certain  wave 


Radio  Instruments  and  Measurements 


47 


lengths  need  to  be  eliminated.  For  example,  the  emf  from  an 
arc  generator  is  not  a  pure  sine  wave  but  contains  harmonics  in 
addition  to  the  fundamental  frequency.  Some  harmonic  may 
be  especially  strong  and  it  may  be  desired  to  suppress  it.  This 
can  be  accomplished  in  some  cases  by  connecting  a  condenser 
across  a  loading  coil  (which  is  not  a  coil  used  to  introduce  the  emf 
into  the  circuit)  either  in  the  closed  circuit  or  the  antenna  and 
tuning  the  combination  of  loading  coil  and  condenser  to  the  ob- 
jectionable frequency.  Various  modifications  of  this  simple 
scheme  can  be  used,  which  may  be  more  convenient  under  certain 
circumstances.  Thus,  instead  of  a  condenser  only,  a  condenser 
and  coil  in  series  can  be  connected  around  the  main  inductance 
as  in  Fig.  31.  The  circuit  LbMC2  is  independently  tuned  to 


FIG.  31. — Coupled  circuits  involving  two  simple  series  cir- 
cuits; -various  modifications  of  this  circuit  are  used  in  trans- 
mitting sets  in  which  it  is  desired  to  emit  certain  frequencies 
and  suppress  others 

the  harmonic  which  is  to  be  suppressed.  The  main  circuit 
is  then  tuned  to  the  frequency  which  is  to  be  emitted.  The 
reactance  to  the  emitted  frequency  is  thus  made  zero,  while 
the  reactance  to  the  objectionable  frequency  is  made  very 
large,  as  shown  in  Fig.  32.  The  reactance  of  the  parallel 
combination  of  M  with  Lb  and  C2  is  found  by  the  method  used 
before  to  be  the  curve  X."  with  two  branches.  The  condenser 

reactance -£-  is  added  to  this,  giving  the  heavy  curve  X  of  react- 
ance to  current  flowing  through  the  ammeter. 

One  of  the  characteristic  properties  of  coupled  circuits  is 
brought  out  by  Fig.  32,  viz,  the  reactance  is  zero  at  two  frequen- 
cies. That  is  the  current  is  a  maximum  for  two  different  fre- 

35601°— 18 4 


48 


Circular  of  the  Bureau  of  Standards 


quencies.  Between  these  two,  indicated  by  a/  and  co",  is  the 
frequency  of  infinite  reactance  or  minimum  current,  indicated  by 
co0.  Thus,  it  is  possible  to  suppress  a  certain  frequency  and  tune 
the  circuit  to  a  different  frequency  either  larger  or  smaller  than 
the  one  suppressed.  There  will  be  current  maxima  at  both  fre- 
quencies (corresponding  to  w'  and  «")  in  the  circuit  if  the  source 
of  emf  supplies  these  two  frequencies  simultaneously.  The  current 
will  have  a  minimum  at  the  intermediate  frequency  of  infinite 
reactance  only  provided  the  resistances  of  the  circuits  are  small. 


JOOO 


FlG.  32. — Reactance  diagram  for  the  simple  coupled  circuits  of  Fig.  31;  the  curve  X  of 
resultant  reactance  is  zero  for  two  values  of  frequency 

IS.  KINDS  OF  COUPLING 

Circuits  may  be  connected  or  coupled  together  in  a  number 
of  ways.  When  there  are  two  circuits,  the  one  containing 
the  source  of  power  is  called  the  primary,  the  other  circuit  the 
secondary.  These  are  generally  coupled  in  one  of  the  follow- 
ing ways:  (a)  By  direct  connection  across  an  inductance  coil; 
(6)  by  electro-magnetic  induction;  (c)  by  direct  connection 
across  a  condenser.  In  the  first  kind,  called  "direct  coupling," 
an  inductance  coil  is  common  to  the  two  circuits  as  illus- 
trated in  Fig.  33  (a).  In  the  second  kind,  "inductive  coup- 
ling," shown  in  (6),  the  two  circuits  are  connected  only  by  mu- 
tual inductance.  An  example  of  the  third  kind,  "capacitive 


Radio  Instruments  and  Measurements 


49 


coupling,"  is  shown  in  (c);   a  condenser  is  common  to  both  cir- 
cuits in  place  of  the  coil  M  of  Fig.  33  (a). 

It  is  characteristic  of  coupled  circuits  that  the  impedances  in 
each  circuit  affect  the  current  flowing  in  the  other.  This  reaction 
of  the  circuits  upon  each  other  is  the  more  marked  when  the 
common  portion  of  the  two  circuits  is  a  larger  proportion  of  their 
impedances.  When  this  is  large  the  coupling  is  said  to  be  "close" 
and  when  small  the  coupling  is  "loose."  In  the  case  of  extremely 
loose  coupling,  the  back  action  of  the  secondary  on  the  primary 
circuit  is  negligible,  and  the  considerations  of  coupled  circuits  do 
not  apply ;  the  two  circuits  act  practically  as  independent  circuits, 
the  primary  merely  applying  an  electromotive  force  to  the 
secondary. 


(c) 


FIG.  33.  —  Types  of  coupling;  (a)  direct  coupling,  (b)  inductive  coupling,  (c)  capacitive 

coupling 

Coupling  Coefficient.  —  The  closeness  of  coupling  is  specified  by  a 
quantity  called  the  coupling  coefficient.     This  is  defined  as  the 


,   where   Xm   is   the  mutual  or  common  reactance  \ 


ratio 


(either  inductive  or  capacitive)  and  X^  is  the  total  inductive  or 
capacitive  reactance  in  the  primary  circuit  and  X2  the  total 
similar  reactance  in  the  secondary.  Thus,  in  the  case  of  direct 
coupling,  Fig.  33  (a),  the  coupling  coefficient  is 

coM 


V«(La+M)co(Lb+M) 

Denote  the  total  inductance  of  primary  and  secondary  by  Lt  and  L2, 
respectively,  and  the  coupling  coefficient  by  k;  then 


k 


M 


50  Circular  of  the  Bureau  of  Standards 

This  also  gives  the  coupling  coefficient  for  inductively  coupled 
circuits,  as  illustrated  in  Fig.  33  (6) ,  Ll  and  L2  being  the  respective 
total  inductances  of  primary  and  secondary,  each  measured  with 
the  other  circuit  removed.  As  suggested  by  the  identity  of 
expression  for  coupling  coefficient,  inductively  coupled  circuits 
may  be  considered  as  equivalent  to  direct-coupled  circuits  having 
the  same  M,  Clt  and  C2,  and  in  which  LSL  =  L1  —  M  and  Lb  =L,  —  M. 
The  coupling  coefficient  in  Fig.  33  (c)  is: 


coCr 


l-+-L-} 

Cb    wCm/ 


CoCb 

Denote  by  C\  the  total  capacity  of  the  primary  circuit,  and  by  C3 
the  total  capacity  in  the  secondary. 


ii        i        A  i       i        i 

r  =  T=T  +  =r~  and.     r  —  -        r  ~~ 


whence, 


c 


L, 


Cb 


L 


FIG.  34. — Special  case  of  capacilive  coupling 

From  these  expressions  the  coupling  coefficient  may  be  obtained 
for  particular  cases.  Thus,  for  Fig.  31,  which  is  a  special  case  of 
the  kind  of  coupling  shown  in  Fig.  33  (a) , 


M 


Lb+M 
Similarly,  Fig.  34  shows   a   special    case  of  capacitive  coupling. 


Radio  Instruments  and  Measurements 
The  coupling  coefficient  is  readily  found  to  be 


V* 


+cm 

Use  of  Coupled  Circuits  to  Select  Frequencies. — Any  of  the 
systems  of  coupled  circuits  which  have  been  mentioned  may  be 
used  for  the  purpose  of  suppressing  current  of  one  frequency  while 
responding  to  current  of  another  frequency  or  wave  length.  This 
was  discussed  above  in  connection  with  the  simple  case  of  direct 
coupling  in  Fig.  29.  It  may  be  shown  that  each  of  the  more 
general  circuits  in  Fig.  33  will  accomplish  the  same  thing.  This 


FIG.  35. — Reactance  diagram  for  case  of  capacitive  coupling  shown  in  Fig.  34 

is  also  true  of  the  simple  case  of  capacitive  coupling  in  Fig.  34, 
as  may  be  seen  from  its  reactance  diagram  Fig.  35.  The  curve 
X"  gives  the  reactance  of  the  parallel  combination  of  Cm  with 
L2  and  Ct>.  Adding  the  reactance  of  Lt  to  this,  the  heavy  curve 
X  is  obtained,  showing  the  total  reactance  to  current  in  the 
primary  circuit.  As  before,  the  reactance  is  zero  at  two  fre- 
quencies and  is  infinite  at  one  intermediate  frequency. 

Thus  any  of  these  arrangements  of  coupled  circuits  may  be 
used  to  remove  an  objectionable  frequency  while  tuning  to  some 
other  frequency  either  higher  or  lower  than  the  one  suppressed, 
provided  the  resistances  of  the  circuits  are  not  large.  In  every 


52  Circular  of  the  Bureau  of  Standards 

case,  co0  corresponding  to  the  frequency  suppressed  in  the  primary 
circuit  is  given  by 

i 


where  L2  =  total  inductance  of  the  secondary  circuit  and  C2  =  total 
capacity. 

16.  DIRECT  COUPLING 

The  above  discussion  shows  how  a  qualitative  comprehension  of 
the  action  of  coupled  circuits  may  readily  be  obtained.     The  exact 

frequencies  to  which  a  coupled 
system  responds  may  be  obtained 
by  calculation  in  the  manner  here 
shown  for  direct  coupling,  upon 
the  assumption  that  resistances 
can  be  neglected. 

The  emf  E  in  the  primary  (Fig. 
36)  is  opposed  by  the  impedance 
^G.36.-CucuitsinWMngdirectcoupling  of  La^fC,,  and  of  the  parallel  com- 
bination of  M  with  Lb  and  C2. 

Denoting  by  1^  and  /2  the  currents  in  primary  and  secondary,  re- 
spectively, the  current  in  M  is  1^  - I2. 


>-/,) 


(32) 


The  emf  across  M  is  the  same  as  that  across  Lb  and  C2  in  series, 
hence 


Therefore, 


E  i 

T  =coLa ^ 


(33) 


The  last  term  is  X" ',  the  reactance  of  the  parallel  combination  of 
M  with  Lb  and  C2,  which  may  be  shown,  as  before,  to  vary  with 


Radio  Instruments  and  Measurements  53 

frequency  according  to  the  curve  marked  X"  in  Fig.  32.     Adding 

to  this  the  curve  of  «La  —  77-.  the  total  reactance  curve  X  is 

coCi 

obtained  (Fig.  37).    This  curve  is  the  graph  of   equation  (33). 

E  . 
The  value  of  co  at  which  j-  is  oo  ,  or  the  current  in  the  primary  a 

M 

minimum,  is  obtained  when  the  denominator  of  the  last  term  is  zero, 
co(Lb+M)  —  7=r=o.     Thus  co2  =    .  —  -  The  symbol  co,  is 


used  to  indicate  that  this  is  the  value  for  resonance  in  the  sec- 
ondary circuit  C2LbM  when  the  primary  circuit  is  open. 

The  values  of  co  at  which  the  primary  current  is  a  maximum 
are  given  by  equating  (33)  to  o  and  solving  for  co.  A  similar  ex- 
pression involving  the  secondary  current  may  be  treated  in  the 
same  way,  and  it  is  found  that  the  secondary  current  has  maxima 
at  the  same  values  of  co  as  the  primary  current.  Expressed  in 
terms  of  the  inductances  and  capacities,  the  solution  is  rather 
complicated.  It  is  more  convenient  to  express  it  in  terms  of  k, 
the  coupling  coefficient,  and  ^  and  co2,  the  respective  values  of 
co  for  resonance  in  the  primary  circuit  CtLaM  alone  and  in  the 
secondary  circuit  C2LbM  alone. 

Using  the  relations, 

,  M  i  i 

(Lb+M)'(        7ZC+H53T-     ~V(Lb+M)C2' 


the  following  two  values  are  found  for  which  the  currents  have 
maxima, 


/ 
V 

=  A 

V 


co,2  +  co22  -     W  -  co22)  f     , 

2(1  -k2) 


w  -  "22  (36) 


—  K  ) 


Example.  —  The  theory  was  experimentally  verified  in  the  fol- 
lowing case.  Two  circuits  were  direct-coupled  as  in  Fig.  36.  The 
following  capacities  and  inductances  were  used  : 

Cl  =0.0023  x  io~6  farad 
C2  =0.00093  X  I0~6  farad 
Lf&  =  56  X  iO"6  henry 
Lb  =  209  X  iQ-6  henry 
M  =  241  X  io~6  henry 


54 


Circular  of  the  Bureau  of  Standards 


The  coupling  coefficient  and  the  values  of  w  for  resonance  in  the 
primary  circuit  alone  and  in  the  secondary  circuit  alone  are  found 
by  (34)  to  be 

£=0.659 

IO6 


The  frequencies  for  maximum  current  in  the  coupled  system  are 
found  by  (35)  and  (36)  to  be 

o/  =  i.o37X  io6 
eo"  =  2.392  X  io6 


&OOOCL 


1OOO 


1O00 


20OO 


FlG.  37. — Reactances  and  current  in  cas£  of  direct  coupling 

It  is  thus  evident  that  the  effect  of  coupling  is  to  spread  out  or 
separate  farther  the  two  independent  frequencies.  Values  of 
primary  reactance  were  calculated  b)^  equation  (33)  for  a  number 
of  values  of  w,  giving  the  curve  X  of  Fig.  37.  It  will  be  noted 
that  it  crosses  the  co  axis  at  the  values  just  given  for  a'  and  co" ',  and 
that  it  goes  to  +  and  —  infinity  at  1.545  X  io6,  the  value  of  co2. 

Current  was  produced  in  the  primary  circuit  by  induction  from 
a  buzzer,  the  buzzer  circuit  being  varied  to  supply  different  fre- 
quencies. The  coil  La  was  inductively  coupled  to  the  buzzer 
circuit,  the  coupling  with  that  circuit  being  so  loose  that  the  emf 
could  be  considered  as  applied  at  one  point  of  the  circuit  L 


Radio  Instruments  and  Measurements  55 

Current  was  measured  by  a  galvanometer  and  a  crystal  detector 
attached  to  a  circuit  inductively  coupled  to  the  secondary  circuit 
Lb  MC2.  The  galvanometer  deflections  were  approximately  pro- 
portional to  the  square  of  the  current.  As  shown,  the  curve  of 
observed  galvanometer  deflections  for  varying  frequency  has  two 
maxima  corresponding  closely  to  co'  and  co".  The  slight  discrep- 
ancies are  probably  'due  to  inaccuracies  in  the  values  used  for  Lj 
and  L2 ;  the  inductances  of  these  coils  were  later  found  to  vary 
slightly  with  frequency,  whereas  a  constant  value  was  assumed 
for  each  in  computing  the  reactance  curve. 

Special  Cases. — In  the  special  case  when  co1=co2,  equations  (35) 
and  (36)  become 

">'=^r  (37) 


(38) 


When  k  is  very  .small;  that  is,  when  M  is  very  small  compared 
with  La  or  Lb, 

co'=co"=cot  (39) 

In  this  case,  where  the  coupling  is  very  loose,  the  system  responds 
to  only  one  frequency  instead  of  two,  and  this  is  the  frequency  of 
resonance  of  either  circuit  by  itself. 

When,  on  the  other  hand,  La  and  Lb  are  very  'small  compared 
with  M,  the  coupling  is  said  to  be  very  close,  and  k  approaches 
the  value  unity.  As  k  increases  to  this  value,  the  two  frequencies 
become  more  widely  separated  and  in  the  limit 

«'=-:=«!  (40) 

-V/2 

co"  =  oo  (41) 

Practically  this  means  that  when  oj1=w2  and  La  and  Lb  are 
negligible  in  comparison  with  M,  there  is  only  one  frequency  and 

this  is  given  by  co'  =   \~wiir'    ^e  reactance  curve  of  such  a  sys- 

" 


tern  is  of  the  type  showTi  in  Fig.  30.  The  curve  X  crosses  the  co 
axis  at  co',  a  value  less  than  ^  (called  co0  in  the  figure),  and 
touches  the  co  axis  again  at  infinity. 

A  particularly  interesting  special  case  of  direct  coupling  is  that 
in  which  co1  =  coz  and  Ll=L2.     This  is  obtained  when  La  =Lb  and 


6  Circular  of  the  Bureau  of  Standards 

\  =  C2.     The  values  of  co  for  maximum  current  in  the  primary  are 

(42) 


co  = 


In  this  case  one  of  the  frequencies  is  constant,  not  varying  when 
Lg.  is  kept  constant  and  M  is  varied.  In  the  reactance  diagram, 
Fig.  37,  the  point  co"  remains  fixed,  and  co'  moves  farther  to  the 
left  as  M  is  increased.  When  M  is  extremely  small,  the  two  fre- 
quencies are  equal,  and  the  equations  (42)  and  (43)  reduce  to  (39). 
When  M  is  very  large  in  comparison  with  La,  the  equations  reduce 
to  (40)  and  (41). 

17.  INDUCTIVE  COUPLING 

The  applied  emf  E  in  the  primary  of  two  inductively  coupled 
circuits  must  satisfy : 


E  = 


,  -  coM/2 


(44) 


The  primary  current  and  reactance  can  be  found  by  writing  down 
a  similar  equation  for  the  secondary  circuit  and  solving.  This  is 
not  necessary,  however,  as  the  solution  already  obtained  for  direct 
coupling  applies  to  this  case  also.  Consider  Lt  to  be  made  up  of 
two  inductances  in  series,  M  and  La,  the  latter  being  given  by 
La^^i  —  M.  Similarly  consider  L2  to  consist  of  two  parts  in 


series,  M  and  Lb=L2-M. 
and  equation  (44)  becomes 


E  = 


Then  Fig   38  is  replaced  by  Fig.  36, 


(45) 


cod 


or, 


-M 


FIG.  38. — Circuits  involving  inductive  coupling 

This  is  the  same  as  equation  (32)  for  direct-coupled  circuits. 
The  two  cases  are,  therefore,  equivalent. 


Radio  Instruments  and  Measurements  57 

Equivalent  Direct  Coupling. — Thus,  an  inductively  coupled 
system  may  be  considered  to  be  replaced  by  the  direct-coupled 
system  of  Fig.  36,  in  which 


Lb=L2-M 

The  reactance  curves  are  the  same,  and  the  frequencies  of  maxi- 
mum current,  given  by  «'  and  w",  (35)  and  (36),  are  the  same. 
Equations  (34)  are  more  convenient  in  the  following  form: 

M  i  i 

=»  w, : 


The  example  given  in  Fig.  37  was  actually  for  direct-coupled 
circuits,  but  corresponds  also  to  a  case  of  inductively  coupled 
circuits  in  which 

L!  =  297  X  io~6  henry, 

L2  =  450  x  io-6  henry, 

and  Clt  C2,  and  M  are  the  same  as  before. 

The  special  cases  treated  above,  in  which  co1  =  co2  may  be  con- 
sidered as  special  cases  of  inductive  coupling  as  well  as  of  direct 
coupling,  except  that  the  last  case,  where  1^  =  1^,  is  of  no  par- 
ticular interest  when  the  coupling  is  inductive,  because  when  M 
is  varied  La  is  not  usually  kept  constant.  With  inductive  coup- 
ling, M  is  usually  varied  by  moving  the  coils  with  reference  to 
one  another,  Lx  and  L2  remaining  constant. 

Example.  —  A  test  of  this  theory  of  inductively  coupled  circuits 
was  made  by  a  set  of  measurements  upon  two  circuits  arranged 
as  in  Fig.  38.  The  coupling  was  varied  by  changing  the  distance 
apart  of  the  two  coils  Lt  and  L2.  The  effect  of  varying  coupling 
is  shown  in  Fig.  39.  As  the  coils  are  brought  closer  together, 
increasing  the  coupling,  the  resonance  points  co'  and  co"  be- 
come more  widely  separated.  The  constants  were  as  follows: 


Cv  =  0.000244  microfarad, 
C2  =  0.000098  microfarad, 
=  103.5  microhenries 


!  .  , 

L2  ==  246.9  microhenries, 
M  =  o.6,  2.0,  5.1,  25.0  microhenries,  successively. 


58  Circular  of  the  Bureau  of  Standards 

The  reactance  curves  were  calculated  from  these  data  and  the 
preceding  formulas.  The  curves  of  current  squared  were  plotted 
from  observations  of  deflections  of  a  galvanometer  connected 


20 


FIG.  39. — Effect  of  "varying  the  coupling  upon  reactance  and  resonance  curves  for  induc- 
tively coupled  circuits 

to  a  thermocouple  loosely  coupled  to  the  secondary  circuit,  as  a 
function  of  the  frequency  of  the  current  which  was  induced  in 
the  primary  by  coupling  loosely  to  a  pliotron  circuit.  Each 


Radio  Instruments  and  Measurements 


59 


mutual  inductance  was  measured  by  two  measurements  of  the 
self -inductance  of  the  two  coils  connected  in  series,  the  connections 
of  one  coil  being  reversed  for  the  second  measurement.  While 
there  are  slight  discrepancies  in  the  agreement  between  the 
points  of  zero  reactance  and  maximum  current,  due  to  slight 
changes  of  the  inductances  with  frequency,  the  agreement  is 
considered  very  good. 

Effect  of  Coupling  on  Currents. — To  calculate  the  current  in  the 
coupled  circuits  requires  that  account  be  taken  of  the  resistances. 
A  specially  important  case  is  that  in  which  the  primary  and 
secondary  circuits  are  both  tuned,  so  as  to  be  separately  in 
resonance  with  the  applied  electromotive  force, 


03  = 


^cTvfe  (46) 

Letting  ./?!  =  resistance  of  primary  circuit  and  R2  —  resistance  of 
secondary,  it  may  be  shown  4  that 


FIG.  40.  —  Variation  of  current  72  -with  coupling  in 
tuned  circuits  inductively  coupled 

M  in  these  formulas  is  supposed  to  be  in  henries.     For  varying 
values  of  M  the  current  in  the  secondary  is  a  maximum  when 


This  also  holds  for  maximum  current  for  a  variation  of  co,  pro- 
vided the  relations  (46)  are  maintained  by  variations  of  the 
capacities.  Other  cases  of  this  sort  are  solved  in  the  reference 
cited  below.4 

4  See  reference  Nos.  15  and  24,  Appendix  a. 


6o 


Circular  of  the  Bureau  of  Standards 
18.  CAPACITIVE  COUPLING 


The  phenomena  in  a  pair  of  coupled  circuits  joined  by  capaci- 
tive  coupling  may  be  shown  in  a  manner  similar  to  the  above 
discussion  of  difect  coupling.  Denoting  by  I±  and  72  the  currents 
in  the  primary  and  secondary,  respectively,  the  current  in  Cm  is 
7X  — 72,  and 


coL, 


cod 


Therefore, 


E 

/ 
I 


i 

<i  ~~      ,~i 
0)Ca 


. 

coL 


II 


(47) 


Cb 


I, 


la 


FIG.  41. — Circuits  involving  capacitive  coupling 

This  is  the  expression  for  reactance  to  current  in  the  primary 
circuit.     A  curve  of  its  variation  with  frequency  is  shown  in  Fig. . 

42.     The  total  reactance  is  X,  the  sum  of  ( coZ^ — ^r  j  and  X", 

the  last  term  in  (47) . 

The  values  of  a/  and  <o"  at  which  the  currents  have  maxima 
are  readily  found  by  equating  (47)  to  o  and  solving  for  w.  It  is 
convenient  to  express  them  in  terms  of  the  coupling  coefficient 
and  the  respective  values  of  co  for  resonance  in  the  primary  cir- 
cuit L!  Ca  Cm  alone  and  in  the  secondary  circuit  L2  Cb  Cm  alone. 
Using  the  relations, 


(cb+cmy 


ca+cr 
c  c 

I*-"  a*—  i 


W,  = 


Radio  Instruments  and  Measurements 


61 


it  turns  out  that  the  currents  have  maxima  at  the  two  frequen- 
cies given  by 


cy  +  co22  +  V  (cot2  -  <Q22) 2  +  4  feX  W 

2 

coi2  +  co22  -  V  (cot2  -  co22) 2  +  4  feX W 
2 


(48) 

(49) 


JQOO 


^«w 


^25C 


FIG.  42. — Reactance  diagram  for  capacitive  coupling  shown  in  Fig.  41 

Special  Cases. — When  w1=co2,  these  expressions  simplify  to 

a/  =co!-\/i  +k 
u>"  =0)^1  —k 
When  the  coupling  is  very  loose,  k  approaches  o,  and 


The  system  responds  simply  to  the  frequency  of  resonance  of 
either  circuit  by  itself.  When,  on  the  other  hand,  Cm  is  small  in 
comparison  with  Ca  and  d,,  the  coupling  is  close  and  in  the  limit 
(48)  and  (49)  reduce  to 


co=o 


62  Circular  of  the  Bureau  of  Standards 

Practically  this  means  that  the  system  responds  to  only  one  fre- 
quency,  given   by   <*>'  =\  T  r  .     It  should  be  noted  that  <ar   is 

V  -t^iLm 

greater  than  <a1}  while  in  the  similar  case  of  direct  coupling  co'  is 
less  than  cox. 

When  u>l=u2  and  Lt=L2  (of  course  also  Ca  =  Cb), 


As  in  the  similar  case  of  direct  coupling,  one  of  the  frequencies  is 
constant,  not  varying  when  Ca  is  kept  constant  and  Cm  is  varied. 
More  General  Cases. — The  kind  of  capacitive  coupling  treated 
in  the  foregoing  is  a  simple  case  of  the  more  general  type  of  capaci- 
tive coupling  shown  in  Fig.  43.  The  expressions  for  coupling 


1 

3> 

"^ 

I> 

[[  * 

0 

:> 

ii 

0> 

I> 

^ 

^>L.                 cj 

r> 
D 

-•     ha 

z> 

:> 

o> 

r 

FIG.  43. — Generalized  case  of  capacitiue 
coupling 


FIG.  44. — Special  type  of  case  shown  in 
Fiff-  43 


coefficient,  etc.,  which  are  complicated  in  the  general  case,  are 
treated  by  E.  Bellini  (La  Lumiere  Electrique,  32,  p.  241;  1916). 
Another  simple  case  which  has  been  found  useful  is  that  shown 
in  Fig.  44.  For  this  kind  of  capacitive  coupling, 


r 

V(c'+c3)  (C"+c3) 

Here  again,  in  the  special  case  of  Wj  =co2  and  C'  =*C",  one  of  the 
frequencies  is  constant,  not  varying  when  C'  is  kept  constant 
and  C3  is  varied. 

For  further  information  on  coupled  circuits,  calculation  of  the 
currents,  transformation  ratios,  etc.,  the  reader  is  referred  to 
Fleming's  The  Principles  of  Electric  Wave  Telegraphy  and  Te- 
lephony, Chapter  III. 

19.  CAPACITY  OF  INDUCTANCE  COILS 

The  small  capacities  between  the  turns  of  a  coil  are  of  such  im- 
portance in  radio  design  and  measurements  that  a  coil  can  seldom 
be  regarded  as  a  pure  inductance.  The  effect  of  this  distributed 


Radio  Instruments  and  Measurements 


capacity  is  ordinarily  negligible  at  low  frequencies,  but  it  modi- 
fies greatly  the  behavior  of  a  coil  at  radio  frequencies.  For  most 
purposes  a  coil  can  be  considered  as  an  inductance  with  a  small 
capacity  in  parallel  as  shown  in  Fig.  45.  This  fictitious  equiva- 


c< 


FIG.  45. — Circuit  which  is  equivalent  to  a 
coil  having  distributed  capacity 


FIG.  46. — Coil  having  capacity,  with  emf  in 
series;  a  case  of  parallel  resonance 


lent  capacity  is  called  the  capacity  of  the  coil.  Investigations 
have  shown  that  in  ordinary  coils  its  magnitude  does  not  vary 
with  frequency.  Thus  a  coil  may  in  itself  constitute  a  complete 
oscillating  circuit  even  when  the  ends  of  the  coil  are  open. 


fOOOO 


8000 


2.000 




O  S  ~~ZO  15  £0 

FIG.  47. — Reactance  diagram  for  coil  having  capacity  with  emf  in  series 

Emf  in  Series  with  the  Coil. — If  such  a  coil  is  placed  in  a  circuit 
with  an  electromotive  force  in  series,  the  case  is  one  of  parallel 
resonance.  The  reactance  curve  will  be  as  shown  in  Fig.  47, 
which  has  the  same  shape  as  the  left  branch  of  the  resultant  in 

35601°— 18 5 


64 


Circular  of  the  Bureau  of  Standards 


Fig.  28.  The  right  branch  is  of  no  interest  and  is  not  shown 
here,  because  for  higher  frequencies  than  co0  (at  which  the  react- 
ance becomes  infinite)  the  coil  no  longer  functions  as  an  induct- 
ance. If  the  resistance  is  negligible,  the  current  due  to  the  elec- 
tromotive force  E  is 


-  co2C<>L 


—  T- 

coL 


5oo 


400 


900 


200 


JOV 


i 


Meters 


JOO 


£OO  30O  400 

FIG.  48. — Variation  of  apparent  inductance  of  a  coil  with  wave  length 

The  apparent  inductance  of  the  coil,  which  would  be  obtained 
by  measurement  of  the  coil  as  an  inductance,  is  La  in 


L 


Ju 

Comparing  with  the  above  expression,  La  =  T  +^c  L 

When  «2C0L  is  small  compared  with  i ,  this  becomes 

La  =  L(i+co2C0L), 


(50) 


Radio  Instruments  and  Measurements  65 

at  frequencies  remote  from  co0,   and  for  C0  in  farads  and  L  in 
henries. 

It  is  usually  convenient  to  calculate  the  apparent  inductance 
in  terms  of  wave  length.     (See  sec.  78.)     Equation  (51)  becomes 


where  C0  is  in  micromicrofarads,  L  in  microhenries,  and  X  in  meters. 
This  holds  except  for  wave  lengths  near  that  corresponding  to  <o0, 
in  which  case  the  more  accurate  expression  applies: 


L 


i  -3-553  TT 


(52) 


FIG.  49 — Coil  having  capacity  -with  emf 
generated  in  the  coil 

As  X0,  the  wave  length  corresponding  to  «0,  is  approached,  La 
becomes  very  large.  X0  is  the  wave  length  at  which  the  inductance 
and  capacity  of  the  coil  would  be  in  resonance,  the  coil  itself  con- 
stituting a  complete  oscillating  circuit.  This  is  a  good  example 
of  parallel  resonance.  The  wave  length  X0  is  called  the  funda- 
mental wave  length  of  the  coil,  similar  to  the  fundamental  wave 
length  of  an  antenna. 

Emf  Induced  in  Coil. — If  a  condenser  is  connected  across  the 
ends  of  a  coil  and  the  coil  is  loosely  coupled  to  a  source  so  that  an 
electromotive  force  is  induced  in  the  coil,  the  total  capacity  in  the 
circuit  will  be  the  sum  of  the  condenser  capacity  and  the  coil 
capacity.  This  is  shown  in  Fig.  49,  C0  and  Ct  being  in  parallel, 
and  the  induced  emf  being  indicated  by  the  electromotive  force  E. 
If  now  the  inductance  be  calculated  for  any  wave  length  from  the 
capacity  of  the  condenser  C,  which  causes  resonance  at  that  wave 
length,  taking  no  account  of  the  coil  capacity  C0,  the  apparent 


66 


Circular  of  the  Bureau  of  Standards 


inductance    La    so    obtained    will    be    greater    than    the    pure 
inductance  L.     This  is  readily  seen  from 


L 


"04;) 


(53) 


Thus  the  apparent  inductance  becomes  greater  the  smaller  the 
capacity  Cl  connected  to  the  coil  to  produce  resonance — i.  e.,  the 
smaller  the  wave  length  at  which  the  measurement  is  made — just 
as  in  Fig.  48  above. 


FIG.  50. — Effect  of  distributed  capacity 
in  the  unused  turns  of  a  coil 

Formula  (53)  is  identically  equivalent  to  formula  (50)  above. 
Thus  the  apparent  inductance  of  a  coil  varies  with  the  frequency 
in  the  same  manner  whether  the  electromotive  force  is  applied  in 
series  with  the  coil  or  by  induction  in  the  coil  itself.  The  pure 
induction  L  and  the  capacity  C0,  to  which  the  coil  is  equivalent, 
may  be  determined  by  either  of  the  two  methods  given  on  page  1 36 
below.  The  'simple  theory  applies  when  the  resistance  is  negli- 
gible, a  condition  which  is  ordinarily  met  at  radio  frequencies. 

Effects  of  Dead  Ends. — The  capacities  of  coils  frequently  give 
rise  to  peculiar  and  undesirable  effects  in  radio  circuits.  Among 
these  are  the  effects  caused  by  the  capacities  of  those  parts  of  a 
coil  which  are  not  connected  in  the  circuit.  For  example,  in  many 
radio  sets  spiral  coils  with  many  turns  are  used  in  which  the  in- 


Radio  Instruments  and  Measurements 


FIG.  51. — Wavemeter  circuit  having  coil 
•with  taps 


ductance  may  be  varied  by  attaching  a  clip  to  any  turn,  thus  uti- 
lizing more  or  fewer  turns.  The  turns  which  are  supposedly ' ' dead' ' 
may  actually  produce  considerable  effect  upon  the  circuit  both  in 
respect  to  energy  loss  and  the  frequency  of  resonance.  As  in  the 
diagram  (Fig.  50),  suppose  that  a  few  turns  of  the  coil  are  con- 
nected across  a  condenser  C± 
(circuit  i)  and  high  frequency 
oscillations  are  set  up  in  this 
circuit.  The  numerous  over- 
hanging or  unused  turns  are 
in  the  magnetic  field  of  the 
used  turns  and  are  closed  by 
their  capacity  (indicated  by 
dotted  lines) .  Hence  this  sec- 
ond circuit  is  coupled  closely  to  the  first  and  if  the  resonant 
frequency  of  this  circuit  is  near  that  of  the  first,  considerable 
current  will  flow  in  it,  strongly  affecting  the  apparent  resistance 
and  resonant  frequency  of  the  first  circuit.  Indeed,  the  circuit 

i  may,  in  consequence,  respond 
to  two  frequencies,  as  in  previ- 
ously considered  cases  of  coupled 
circuits.  The  case  is  the  same  as 
that  of  Fig.  31 ,  and  the  reactance 
and  current  curves  of  the  system 
would  be  of  the  same  form  as  m 
Fig.  32.  In  this  case  of  many 
overhanging  turns  compared  to 
the  number  in  use,  it  is  generally 
advisable  to  short-circuit  the 
overhanging  turns,  for  then  the 
impedance  of  the  second  circuit 
becomes  so  high  that  little  cur- 

FIG.  52— The  case  of  coupled  circuits  to  rent  will  flow.     When  only  a  few 

turns  are  overhanging  it  is  best  to 


which  Fig.  51  is  equivalent 


leave  them  open,  for  the  overhanging  turns  will  have  a  greater 
impedance  when  open  than  when  short-circuited. 

Another  common  case  in  which  dead  ends  cause  troublesome 
effects  is  shown  in  Fig.  51,  a  common  form  of  circuit  used  in  re- 
ceiving apparatus  and  in  some  forms  of  wavemeters.  L*  is  a 
coupling  coil.  The  main  coil  L,  of  large  inductance  and  consid- 
erable distributed  capacity,  is  divided  into  sections,  one  or  more 


68  Circular  of  the  Bureau  of  Standards 

of  which  may  be  connected  in  circuit  to  allow  adjustment  for 
various  ranges  of  wave  lengths.  The  fictitious  condenser  C0  in 
parallel  with  this  coil  represents  the  effective  capacity  of  the 
coil.  For  short  wave  lengths  only  part  of  the  coil  is  in  circuit, 
the  unused  sections  being,  however,  inductively  related  to  the  part 
in  circuit.  It  will  be  seen  that  this  arrangement  is  really  a  case 
of  two  direct-coupled  circuits,  which  should  respond  to  two  distinct 
frequencies  or  wave  lengths.  The  circuit  may  be  diagrammati- 
cally  represented  by  the  circuit  in  Fig.  52,  which  is  equivalent 
to  Fig.  31  above.  This  circuit  is  resonant  to  two  frequencies,  as 
shown  by  the  resultant  reactance  curve  of  Fig.  32.  As  the  setting 


5400 

£ 


"to — 35     3o     35     55     eo     TO     eo io     ioo     no     Tao     150     w> BO    ieo fro     iso 
S««mq    of  Wavemeter    Condenser 

FIG.  53. — Calibration  curve  of  a  commercial  wave  meter  with  discontinuity  caused  by 

distributed  capacity 

of  the  condenser  d  is  varied,  the  frequencies  to  which  the  circuit 
responds  are  changed.  For  settings  in  the  neighborhood  of  the 
natural  frequency  of  the  circuit  ll'C0  strong  currents  are  obtained 
at  both  the  resonant  frequencies.  For  frequencies  considerably 
more  remote,  the  current  of  only  one  frequency  is  appreciable. 

This  behavior  is  sometimes  experimentally  found  in  wave 
meters.  As  the  setting  of  the  condenser  is  varied,  the  frequencies 
or  wave  lengths  to  which  the  current  responds  vary,  and  in  the 
neighborhood  of  a  certain  wave  length  there  are  two  wave  lengths 
at  which  resonance  occurs  for  every  condenser  setting.  This  is 
shown  in  Fig.  53,  which  is  an  experimentally  obtained  calibration 
curve  of  a  commercial  wave  meter. 


Radio  Instruments  and  Measurements 


20.  THE  SIMPLE  ANTENNA 


69 


Distributed  Capacity  and  Inductance, — The  current  flowing  into 
a  condenser  is  given  by  I  =  EwC,  and  the  voltage  across  an  induct- 
ance is  given  by  E  =  IuL.  Thus  the  current  into  a  condenser 
(voltage  constant)  and  the  voltage  across  an  inductance  (current 
constant)  increase  as  the  frequency  increases.  Both  of  these 
facts  tend  to  make  the  small  capacities  between  different  portions 
of  a  circuit  more  important,  the  higher  the  frequency.  At  low 
frequencies  in  general  the  current  at  different  points  in  a  circuit 
is  the  same,  and  displacement  currents  are  present  only  where 
relatively  large  condensers  have  been  intentionally  inserted  in 
the  circuit.  The  inductance  and  capacity  are  definitely  localized 
or  ' '  lumped. ' '  At  very  high  frequencies,  however,  or  when  the 
dimensions  of  the  circuit  are  comparable  to  the  wave  length,  the 


'j        i>         ii        »i         ' 

'  *\  A  /  • 

(       M      ,»(        '{        / 

•  V    if    *f    >r    Vl  i 
(  X  A  ^  J 

1 

' 

5) 

•_  :                         b-- 



,  —  i 

. 
i                 f&\  

^  

fa 

FlG.  54.— Circuit  representing  distributed  capacity  and  inductance 

capacities  between  different  parts  of  the  circuit  become  impor- 
tant and  the  current  may  vary  appreciably  from  point  to  point 
in  the  circuit.  Some  of  the  current  leaks  away  from  or  onto  the 
conductor  through  the  capacity  to  other  parts  of  the  circuit,  the 
current  through  inductances  in  different  parts  of  the  same  circuit 
will  be  different,  and  hence  their  inductive  effect  will  be  different. 
In  such  a  case  the  equivalent  capacity  and  inductance  of  the 
circuit  will  depend  upon  the  frequency  and  the  separate  condensers 
and  inductances  must  be  considered  with  regard  to  their  position 
in  the  circuit — i.  e.,  one  has  to  deal  with  ' '  distributed ' '  inductance 
and  capacity.  Fig.  54  represents  a  circuit  or  line  of  two  long 
parallel  wires  supplied  with  current  from  a  generator  and  closed 
at  the  far  end.  The  inductance  of  the  wires  and  the  capacity 
between  them  are  represented  by  condensers  and  inductances 
drawn  in  dotted  lines.  A  number  of  ammeters  are,  for  conven- 
ience, supposed  to  be  inserted  at  points  in  the  circuit.  At  a  low 


yo 


Circular  of  the  Bureau  of  Standards 


frequency  very  little  current  will  flow  into  the  condensers  and 
all  of  the  ammeters  will  read  the  same.  If  the  frequency  is  in- 
creased, more  current  will  flow  through  the  condensers  and  the 
ammeter  readings  will  decrease  successively  from  the  generator  to 
the  far  end.  As  a  result  of  the  changed  distribution  of  current 
in  the  line,  the  equivalent  inductance,  capacity,  and  resistance 
of  the  line  will  vary  with  the  frequency. 

Simple  Antenna.  —  The  simplest  form  of  antenna  is  a  single  ver- 
tical wire,  the  lower  end  of  which  is  connected  to  ground.  This 
forms  an  oscillatory  circuit,  the  inductance  is  due  to  the  wire  and 
the  capacity  is  that  between  the  wire  and  the  ground.  Thus  Fig. 
55  shows  diagrammatically  the  capacity  and  inductance  and  the 
flow  of  current  at  an  instant  of  time.  Some  of  the  current  from 
the  wire  is  continually  flowing  off  by  the  capacity  paths  to  ground, 


FlG.  55.  —  Distributed  capacity  and  indue- 
tance  in  a  vertical  wire,  a  simple  form  of 
antenna 


ViS^^^^/^^^W^ 

FlG.  56. — Distribution  of  current 
and  voltage  in  the  simple  antenna 
•when  oscillating  at  its  fundamental 
frequency 


so  that  the  maximum  current  is  flowing  at  the  base  of  the  antenna, 
while  at  the  extreme  top  there  is  no  current  flowing  in  the  wire. 
The  amplitude  of  the  voltage  alternations  is  zero  at  the  ground  and 
is  a  maximum  at  the  top.  The  distribution  of  current  and  volt- 
age in  approximately  sinusoidal  and  is  shown  in  Fig.  56. 
This  represents  the  fundamental  oscillation  of  a  simple  antenna. 
The  length  of  the  wire  is  equal  to  the  distance  from  node  to  loop  or 
is  one-fourth  of  the  wave  length.  It  is  possible  for  the  simple  an- 
tenna to  oscillate  with  other  distributions  of  current  and  voltage,  in 
which,however,  the  top  must  always  be  a  node  for  current  and  the 
bottom  a  node  for  voltage.  Thus  in  Fig.  57  is  shown  the  next 
possible  oscillation.  Here  the  length  of  the  wire  is  three-fourths 
of  a  wave  length.  Hence  the  wave  length  is  one-third  of  the 
fundamental  or  the  frequency  is  three  times  as  great.  Other  pos- 
sible oscillations  have  frequencies  of  five,  seven,  nine,  etc.,  times 
the  fundamental. 

Antenna  with  Large  End  Capacity.  —  Suppose  that  a  number  of 
long  horizontal  wires  are  attached  to  the  top  of  the  vertical  wire 


Radio  Instruments  and  Measurements 


of  the  simple  antenna,  thus  forming  an  inverted  "  L  "  antenna  as  in 
Fig.  58.  In  this  case  only  a  small  proportion  of  the  current  in 
the  vertical  portion  flows  off  to  ground  through  capacity  paths, 
the  main  capacity  flow  taking  place  from  the  horizontal  portion. 
Thus  the  current  throughout  the  vertical  portion  will  be  very 
nearly  constant.  The  total  capacity  will  be  much  larger  than 
that  of  the  simple  antenna  and  the  inductance  likewise  somewhat 
larger,  hence  the  wave  length  will  be  considerably  increased. 

There  are  a  number  of  other  forms  of  antennas  which  also  have 
large  capacity  areas  at  the  top  of  the  vertical  lead,  such  as  the 
"T,"  "umbrella/'  etc. 


FIG.  57. — The  distribution  of  current 
and  -voltage  in  the  simple  antenna 
when  oscillating  in  the  first  harmonic 


FIG.  58. — Form  of  antenna  of  large 
capacity 


21.  ANTENNA  WITH  UNIFORMLY  DISTRIBUTED  CAPACITY  AND 

INDUCTANCE 

The  mathematical  treatment  of  currents  in  circuits  having  dis- 
tributed capacity  and  inductance  are  generally  concerned  with  the 
case  where  these  quantities  are  uniformly  distributed.  Because 
of  end  effects,  this  condition  can  not  be  strictly  realized  except 
with  circuits  of  infinite  length,  such  as  two  parallel  wires,  a  single 
wire  .with  a  concentric  cylindrical  return,  or  a  single  wire  (or  num- 
ber of  parallel  wires),  and  ground  return.  Such  theory  applies 
approximately,  however,  to  the  simple  vertical-wire  antenna  or 
to  the  horizontal  portion  of  an  inverted  "  I/'  antenna.  ,The  fol- 
lowing notation  is  used  in  this  discussion : 

Z=Iength  of  antenna,  CD.  (Fig.  59), 
Lt=inductance  per  unit  length. 
C\=capacity  per  unit  length. 
L0=^L1— inductance  for  uniform  current. 
C0=ZC!=capacity  for  uniform  voltage. 
Lft=low-frequency  inductance  of  antenna=i/3  L0. 
Ca=lovv-frequency  capacity  of  antenna=C0. 
X=reactance  of  antenna. 
XL= reactance  of  loading  coil. 
Xc=reactance  of  series  condenser. 

L=loading  coil. 

C=series  condenser. 


72 


Circular  of  the  Bureau  of  Standards 


In  Fig.  59  an  inverted  "I/'  antenna  is  drawn  to  represent  a 
circuit  with  uniformly  distributed  capacity  and  inductance,  the 
distributed  quantities  being  represented  by  dotted  lines.  The 
resistance  is  assumed  to  be  negligible.  CD  is  the  horizontal  por- 
tion, BE  the  ground,  and  BC  the  lead-in  which  is  supposed  to  be 
free  from  inductance  or  capacity,  excepting  when  a  coil  or  con- 
denser or  both  are  inserted  at  A. 

Low-Frequency  Capacity  and  Inductance  of  Antenna. — It  is 
desirable  to  explain  the  significance  of  the  quantities  Lt  and  Clf  and 
the  quantities  C0  =  /C1  and  L0  =  /L1.  If  the  portion  CD  were  uni- 
formly charged  to  unit  positive  potential,  then  the  charge  on  each 
unit  of  length  of  CD  would  be  numerically  equal  to  Cl  and  the 
total  charge  on  CD  would  be  C0  =  IC1.  The  antenna  would  be 
charged  in  this  way  if  a  constant  or  slowly  alternating  emf  were 
introduced  at  A,  and  hence  the  quantity  C0  =  lCt  is  sometimes 


ii!  i'li'.iiliiiii  'i  1  1; 

vVVVvvvvV 

i  i  i  1  i  !  i  i  r  i 
1  \l  V   <l  tf 

i  V  v  /  V  ' 

7T7T7- 
rVV 

"- 

..        ._ 

"" 

.. 

" 

i 

..u 

r-       —  r" 

in 



m 

FIG.  59. — Antenna  with  uniformly  distrib- 
uted capacity  and  inductance 

called  the  "  static  "  capacity  of  the  antenna.  It  is  also  called  the 
low-frequency  capacity.  Formulas  for  the  calculation  of  the 
static  capacity  are  given  on  pages  237  to  242.  At  high  frequencies 
the  potential  at  all  points  on  CD  is,  however,  not  the  same  at  a 
given  instant,  and  hence,  the  capacity  (called  the  high-frequency 
or  dynamic  capacity)  is  different.  In  the  case  of  the  inductance, 
suppose  that  a  conductor  of  negligible  resistance  and  inductance 
is  connected  from  D  to  E  and  that  an  emf  which  is  constant  or 
slowly  alternating  is  introduced  at  A.  The  current  flow  at  all 
portions  of  the  circuit  CDEB  at  a  given  instant  would  then  be  the 
same  and  the  total  inductance  L0  of  the  circuit  would  be  numeri- 
cally equal  to  the  total  magnetic  flux  of  lines  linked  with  the  cir- 
cuit when  the  current  is  unity.  This  is  the  value  of  inductance 
which  would  be  calculated  from  the  formulas  on  pages  247  to  250 
The  inductance  per  unit  length  Lt  is  L0  divided  by  the  length. 
In  the  actual  antenna,  however,  the  current  flow  can  not  be  the 


Radio  Instruments  and  Measurements  73 

same  at  all  portions  of  the  circuit,  for  the  current  at  the  open  end 
of  the  aerial  must  always  be  zero,  and  hence  L0  =  /Lx  does  not  rep- 
resent the  inductance  of  the  aerial  at  any  frequency.  It  will  now 
be  shown  that  the  low-frequency  inductance  of  the  aerial  is  one- 
third  L0. 

If  an  alternating  voltage  of  frequency  /  (or  co  =  2irf).  is  introduced 
at  A  in  the  actual  antenna,  it  can  be  shown5  that  the  aerial  or 
horizontal  portion  behaves  as  a  reactance, 

X  =  -       £  cot  <alCLt 


or, 

X=  —  t7=r  COt 


/To 
'Vc; 


The  expansion  of  the  cotangent  of  an  angle  into  a  series  is 


i     x 

cotx  = . 

x     i 


the  remaining  terms  being  negligible  if  x  is  small.  Hence,  if  w  is 
small  —  i.  e.,  for  low  frequencies  —  substituting  co^/CoLo  for  x  in 
the  series,  we  have 


_  JL  , 

~~  ~~ 


This  is  the  reactance  of  a  capacity  C0  in  series 

with  an  inductance  —  -,  as  in  Fig.  60;  hence  we 

o 
see  that  the  static  capacity  is  C0  =  ICi  and  the 

L        IL  FIG.  60.-  -Simple  cir- 

low-frequency  inductance  is  —  =  —  -•  cuu  -which  is  equiva- 

**          •*  /cwi  to  the  antenna 

Fundamental    Wave     Length.  —  The     expres-       shown  in  Fig.  59 
sion    for     the     reactance     of    the     aerial 

Zx  =  —  -\/7=r    cot   co-y/CoLo   shows   that   at   low  frequencies  when 
V  ^-o 

the  cotangent  is  positive,  the  reactance  is  negative,  and  hence  the 
capacity  reactance  overbalances  the  inductive.  As  the  frequency 
increases  the  cotangent  becomes  zero,  which  first  occurs  when 

/  -         7T 

UT/C0L0  =  ~  and  the  reactance  is  zero.  The  frequency  at  which 
this  occurs  is  the  fundamental  frequency  of  natural  oscillation  of 

'See  reference  No.  31,  Appendix  2. 


74  Circular  of  the  Bureau  of  Standards 

the  antenna  when  the  vertical  lead-in  is  free  from  inductance  and 
capacity.  This  frequency  is 

f  =  —  =        X 

I        27T       4^/CoLo 

The  fundamental  wave  length  is  given  by 

A  =  7  =  4.c\C0L,0 

where  c  =  velocity  of  propagation  of  electromagnetic  waves,  and 
in  the  last  equation  Xm  stands  for  wave  length  in  meters,  C0  is  in 
microfarads,  and  L0  in  microhenries. 


ZSoo 


FIG.  61. — Reactance  curve  for  antenna  with  uniformly  distributed  capacity  and  inductance 
(Z  on  curves  corresponds  to  X  in  text) 

Harmonics. — For  frequencies  above  the  fundamental  the  react- 
ance becomes  positive  and  the  inductive  reactance  preponderates 

i 


up  to  the  frequency  /  =  • 


at  which  the  reactance  becomes 


2VCoLc 

infinite.  Beyond  this  frequency  the  reactance  is  again  negative ,  but 
decreases  numerically  with  increasing  frequency  until  it  again 
becomes  zero  and  there  is  a  harmonic  natural  oscillation  at  a 
frequency 

t          3 

4-\/CojLo 


Radio  Instruments  and  Measurements  75 

This  variation  of  the  aerial  reactance  with  the  frequency  is  shown 
by  the  cotangent  curves  in  Fig.  61.  Those  frequencies  for  which 
the  reactance  of  the  aerial  becomes  zero  and  which  are  the  natural 
frequencies  of  oscillation  of  the  antenna  when  the  lead-in  is  of 
zero  reactance  are  given  in  the  figure  by  the  points  of  intersection 
of  the  cotangent  curves  with  the  axis  of  ordinates.  For  these 
points 

m 


X  =  —  -\C0L,0. 
m 

Now  it  is  shown  in  the  theory  of  long  circuits  with  uniformly  dis- 
tributed quantities,  such  as  those  considered  here,  that  at  high 


frequencies  -^C^^—,  approximately,  and  since 
c 

it  follows  that 

me 


m 

Hence  the  wave  lengths  of  the  fundamental  and  harmonic  oscilla- 
tions of  the  antenna  are  approximately  4/,  4/3/,  4/5/,  etc.,  or,  as 
stated  above  in  the  description  of  the  oscillations  of  a  simple 
vertical  antenna,  the  length  of  the  wire  is  approximately  1/4,  3/4, 
5/4,  etc.,  times  the  wave  length.  The  distributions  of  current  and 
voltage  corresponding  to  these  different  possible  modes  of  oscilla- 
tion have  also  been  shown.  These  relations  between  antenna 
length  and  wave  length  are  only  rough  approximations,  because 
of  the  finite  length  of  the  antenna  and  because  the  vertical  portion 
of  the  antenna  has  been  neglected. 

22.  LOADED  ANTENNA 

In  general,  when  an  antenna  is  used  for  transmission  or  recep- 
tion, coils  or  condensers  or  both  are  inserted  in  the  lead-in  to  modify 
the  natural  frequency  of  oscillation  of  the  system;  i.  e.,  to  tune 
it  to  a  given  frequency  or  wave  length.  When  a  reactance  XK  is 
present  in  the  lead-in,  the  natural  frequency  of  oscillation  is  then 
determined  by  the  condition  that  the  total  reactance  of  lead-in 
plus  aerial  shall  be  zero  —  that  is,  XK+X  =  o.  In  the  succeeding 


76 


Circular  of  the  Bureau  of  Standards 


sections  the  cases  will  be  considered  where  an  inductance  or  a 
condenser  is  inserted  in  the  lead-in. 

Antenna  with  Series  Inductance. — If  an  inductance  L0  is  inserted 
in  the  lead-in,  its  reactance  XL  is  equal  to  wL.  This  is  a  positive 
reactance  which  increases  linearly  with  the  frequency  and  is 
represented  in  Fig.  62  by  a  straight  line.  The  reactance  of  the 
aerial  X  is  shown  by  the  cotangent  curves.  The  sum  XL  +  X  is 
drawn  in  heavy  solid  lines.  Those  frequencies  at  which  these 
latter  curves  cut  the  axis  of  abscissas  are  the  frequencies  for  which 


FIG.  62. — Reactance  curve  for  antenna  -with  series  inductance.     (Z  on  curves  corresponds 

to  X  in  text) 

XL  +  X  =  o  and  are  the  natural  frequencies  of  oscillation  of  the 
system.  It  will  be  noted  that  the  insertion  of  the  inductance 
coil  has  decreased  the  natural  frequencies  of  oscillation — i.  e., 
increased  the  wave  length.  Also  the  harmonic  frequencies  are 
no  longer  integral  multiples  of  the  fundamental  as  in  the  case  of 
the  simple  antenna. 

The  condition  XL  +  X  =  o,  which  determines  the  natural  frequen- 
cies of  oscillation,  leads  to  the  equation 


or 


coL  —  -\l~~- 


cot 


L 
To 


Radio  Instruments  and  Measurements  77 

I  c 

Since    VCoL0=-;  o)  =  2irf  and  A  =^  this  equation  may   also   be 

c  I 

written 


27T/       -L0 

~x~ 

These  transcendental  equations  which  determine  co  and  X  can  not 
be  solved  directly;  it  is,  however,  possible  to  solve  them  graphi- 
cally as  shown  in  Fig.  62  or  to  determine  indirectly  a  table  (second 

column  of  Table  i)  which  will  give  the  values  of  a>VCoL0  or  ~v~  for 
different  values  of  y-»  from  which  then  co,  /,  or  X  may  be  determined. 

-i^o 

Table  i  gives  only  the  lowest  value  of  co-v/C0L0  or  TT-,  corresponding 

A 

to  the  fundamental  oscillation  of  the  loaded  antenna.  In  any 
actual  antenna  the  wave  length  would  be  greater  than  that  given 
by  this  calculation  because  of  the  inductance  and  capacity  of  the 
vertical  portion;  this  discussion  deals  only  with  the  horizontal 
portion. 

As  an  example  of  the  method  let  us  assume  the  quantities  used 
in  Figs.  6  1  and  62.  Let  the  length  of  the  antenna  be  /  =  60  meters 
and  the  static  capacity  C0  =  ICV  =  0.0008  microfarad.  Then  since 

I         60 

»  L0  =  50  microhenries. 
8 


c     3  X  io8 

In  the  case  of  the  unloaded  antenna,  the  natural  wave  lengths 
would  be  4/,  4/3/,  etc.  —  that  is,  240,  80,  etc.,  meters;  the  natural 

(c  ==  "^  x  io^\ 
/=  x  meters/  WOuld   be   1-25><lo6»  3-75  X  io8,  etc., 

cycles  per  second;  the  periodicities  (w==2ir/)  7.85  X  io6,  23.6  X  ioa, 
etc.,  radians  per  second  agreeing  with  the  values  in  Fig.  62. 
If  now  an  inductance  L  =  100  microhenries  is  introduced  in  the 

lead-in,  we  have  7-  =  2.     From  Table  i  we  find  that 

•L>o 

_        27T/ 

co  VCoLo  =  -y  =  0.653, 

hence 

0.653 
co=    ,  "^ 

Vo.oooS  X  io-6  X  50  X  io-6 

and 


x 

X  =  —^  —  =  577  meters. 
0-653 


78  Circular  of  the  Bureau  of  Standards 

This  corresponds  to  the  lowest  frequency  of  oscillation  as  shown 
in  Fig.  62.  Introducing  the  inductance  has  increased  the  wave 
length  of  this  oscillation  from  240  to  577  meters. 

The  harmonic  oscillations  are  of  importance  in  some  cases. 
If  the  emf  which  is  applied  to  the  antenna  has  the  fundamental 
frequency  of  the  loaded  antenna,  the  oscillations  will  be  of  that 
frequency  alone.  If,  however,  the  antenna  is  first  charged  and 
then  set  into  oscillation  by  the  breaking  down  of  a  spark  gap  in 
the  antenna  as  in  the  original  Marconi  antenna,  frequencies  corre- 
sponding to  all  of  the  possible  modes  of  oscillation  will  be  emitted 


FIG.  63. — Reactance  curve  for  antenna  -with  series  condenser  (Z  on  curves  corresponds  to 

X  in  text) 

by  the  antenna.  Or  in  the  case  of  the  arc,  which  in  itself  generates 
fundamental  and  harmonic  frequencies,  if  a  harmonic  of  the  arc 
coincides  somewhat  closely  with  one  of  the  harmonic  modes  of 
vibration  of  the  antenna,  this  oscillation  will  be  strongly  rein- 
forced, a  large  amount  of  energy  will  be  wasted,  and  interference 
will  be  caused. 

Antenna  with  Series  Condenser. — Though  not  as  important 
practically  as  the  case  just  considered,  there  are  occasions  when  a 
condenser  is  inserted  in  the  lead-in  of  an  antenna  to  shorten  the 

wave  length.     If  its  capacity  is  C,  its  reactance  will  be  Xc  =  — ~0 

In  Fig.  63  the  cotangent  curves  representing  the  aerial  reactance 
X.  are  again  shown  as  is  also  the  parabola  representing  Xc.  The 


Radio  Instruments  and  Measurements  79 

sum  Xc  +  X  is  drawn  in  heavy  solid  lines  and  crosses  the  axis  at 
points  corresponding  to  the  natural  frequencies  of  oscillation  of 
the  loaded  antenna.  It  will  be  noted  that  the  frequencies  of  oscil- 
lation are  increased — i.  e.,  the  wave  length  shortened — by  the 
insertion  of  the  condenser  and  that  the  harmonic  oscillations  are 
not  integral  multiples  of  the  fundamental.  Fig.  63  shows  that 
o)  is  increased  from  7.85X10*  to  1 0.96X10*  or  the  wave  length 
decreased  from  240  to  172  meters  by  the  insertion  of  a  0.0005 
microfarad  condenser. 

Simple  Calculation  of  the  Wave  Length  of  a  Loaded  Antenna. — 
The  ordinary  formula  for  the  frequency  of  oscillation  of  circuits 
with  lumped  inductance  and  capacity  may  be  applied  to  the 
antenna  with  distributed  constants  in  the  case  of  an  inductance 
coil  in  the  lead-in,  and  the  error  in  computing  the  frequency  or 
wave  length  will  be  small.  The  inductance  and  capacity  of  the 
aerial  at  all  frequencies  is  supposed  to  be  the  same  as  the  low- 
frequency  values;  i.  e.,  --  and  C0.  If  the  loading  coil  has  an 

O 

inductance  L,  the  total  inductance  will  be  L  -f— •     Hence 

o 

'  I 


the  frequency  is 


and  the  wave  length 


where  in  the  last  equation  Xm  means  wave  length  in  meters, 
inductance  is  expressed  in  microhenries,  and  the  capacity  in 
microfarads. 

Applying  this  to  the  numerical  example  worked  out  above  by 
the  exact  theory,  in  which 


*-*o    \    s~*  /        \ 

(55) 


(  L  +  -  '°  J  =  1  1  6.67  microhenries 
C0  =0.0008  microfarads 

35601°—  18  -  6 


8o  Circular  of  the  Bureau  of  Standards 

we  have  A  =  575  meters,  which  differs  only  one-third  of  i  per  cent 
from  the  value  A  =  577  obtained  before. 

The  magnitude  of  the  errors  in  using  the  simple  formula 


-1C, 


is  also  shown  in  Table  i.     In  the  second  column,  as  pointed  out 
before,  are  the  values  of  <aJc<>L0  or  -*—  f°r  different  values  of  y- 

A  •  Z-*o 

as  computed  from  the  exact  cotangent   formula.     The  simple 
formula  gives 


>     3 

and  in  the  third  column  are  given  for  comparison  the  values  of 
w-iJCoLo  computed  on  this  basis.  These  values  are  too  high;  i.  e., 
result  in  too  high  a  value  for  the  frequency  or  too  low  a  value  for 
the  wave  length.  The  per  cent  error  is  given  in  the  last  column. 
The  maximum  error  is  10  per  cent  for  L=o — i.  e.,  at  the  funda- 
mental of  the  antenna — but  the  error  rapidly  decreases  as  L 
increases  and  is  less  than  i  per  cent  for  L  equal  to  or  greater 
than  L0. 

It  has  been  stated  in  several  publications  that  very  large  errors 
would  result  from  applying  the  ordinary  theory  of  circuits  with 
lumped  constants  to  the  case  of  an  antenna  which  has  distributed 
quantities.  This  misconception  has  arisen  because  the  quantity 
Lo  which  occurs  in  the  formula  for  the  distributed  case  was  used 
for  the  inductance  of  the  aerial  in  applying  the  formula  for  the 
case  of  lumped  constants.  We  have  pointed  out  that  L0  could 
not  be  the  inductance  of  the  aerial  at  any  frequency.  When 

— ,  instead  of  L0,  is  used  the  agreement  is  very  close.     In  fact, 

o 

since  this  error  is  usually  less  than  i  per  cent,  it  is  practically 
never  worth  while  to  use  formulas  based  on  the  precise  theory  to 
calculate  wave  length,  because  of  the  uncertainty  introduced  by 
the  vertical  portion  of  the  antenna.  Equation  (55)  is  therefore 
sufficient  for  all  ordinary  calculations. 


Radio  Instruments  and  Measurements 
TABLE  1. — Data  for  Loaded  Antenna  Calculations 


81 


1 

Differ- 

1 

Differ- 

L 
Lo 

uVCoLo 

VcH 

ence,  per 
cent 

L 
Lo 

"VCoLo 

lL+± 
~\Lo3 

ence,  per 
cent 

0.0 
.1 
.2 
.3 
.4 
.5 
.6 
.7 
.8 
.9 
1.0 
1.1 

1.571 
1.429 
1.314 
1.220 
1.142 
1.077 
1.021 
0.973 
.931 
.894 
.860 
.831 

1.732 
1.519 
1.369 
1.257 
1.168 
1.095 
1.035 
0.984 
.939 
.900 
%.866 
'.835 

10.3 
6.3 
4.2 
3.0 
2.3 
1.7 
1.4 
1.1 
0.9 
.7 
.7 
.5 
5 

3.1 
3.2 
3.3 
3.4 
3.5 
3.6 
3.7 
3.8 
3.9 
4.0 
4.5 
5.0 
5  5 

0.539 
.532 
.524 
.517 
.510 
.504 
.4977 
.4916 
.4859 
.4801 
.4548 
.4330 
4141 

0.540 
.532 
.525 
.518 
.511 
.504 
.4979 
.4919 
.4860 
.4804 
.4549 
.4330 
4141 

0.1 
.1 
.1 
.1 
.1 
.0 
.0 
.0 
.0 
.0 
.0 
.0 

1  3 

779 

782 

4 

6  0 

3974 

.3974 

1  4 

757 

760 

4 

6  5 

3826 

.3826 

1    e 

736 

739 

4 

7  0 

3693 

.3693 

1  6 

717 

719 

3 

7  5 

3574 

.3574 

'  1    7 

699 

701 

3 

8  0 

3465 

.3465 

1  8 

683 

685 

3 

8  5 

3366 

.3366 

1  9 

668 

689 

3 

9  o 

3275 

.3275 

2  Q 

653 

655 

3 

9  5 

3189 

3189 

2  1 

640 

641 

2 

10  0 

3111 

.3111 

2  2 

627 

628 

2 

11  0 

2972 

2972 

2  3 

615 

616 

2 

12  0 

2850 

.2850 

2  4 

604 

605 

2 

13  0 

2741 

.2741 

2  5 

593 

594 

2 

14  0 

2644 

.2644 

2  6 

583 

584 

2 

15  0 

2556 

2556 

2  7 

574 

574 

2 

16  0 

2476 

2476 

2  g 

564 

565 

1 

17  0 

2402 

2402 

2  9 

556 

556 

1 

18  0 

2338 

.2338 

3  0 

547 

548 

1 

19  0 

2277 

2277 

20  0 

2219 

2219 

23.  ANTENNA  CONSTANTS 

Antenna  Resistance. — The  power  supplied  to  maintain  oscilla- 
tions in  an  antenna  is  dissipated  in  three  ways:  (i)  Radiation; 
(2)  heat,  due  to  conductor  resistance;  (3)  heat,  due  to  dielectric 
absorption.  (At  high  voltages  there  is  a  further  power  loss  due 
to  brush  discharge;  this  will  not  be  considered  in  the  following.) 
The  first  of  these  represents  the  only  useful  dissipation  of  power 
since  it  is  the  power  which  travels  out  from  the  antenna  in  the 
form  of  the  electromagnetic  waves  which  transmit  the  radio 
signals.  The  amount  of  power  radiated  depends  upon  the  form 
of  the  antenna,  is  proportional  to  the  square  of  the  current  flowing 
at  the  current  antinode  of  the  antenna,  and  inversely  proportional 
to  the  square  of  the  wave  length  of  the  oscillation.  Since  the 


82 


Circular  of  the  Bureau  of  Standards 


dissipation  of  power  is  proportional  to  the  square  of  the  current,  it 
may  be  considered  to  be  caused  by  an  equivalent  or  effective 
resistance,  which  is  called  the  radiation  resistance  of  an  antenna. 
Thus  the  radiation  resistance  of  an  antenna  is  that  resistance 
which,  if  inserted  at  the  antinode  of  current  in  the  antenna  would 
dissipate  the  same  power  as  that  radiated  by  the  antenna.  The 
radiation  resistance  varies  with  the  wave  length  in  the  same  way 
as  the  radiated  power;  i.  e.,  inversely  as  the  square  of  the  wave 
length.  Curve  i  of  Fig.  64  represents  the  variation  of  this  com- 
ponent of  the  resistance  of  an  antenna. 


FlG.  64. — Variation  of  antenna  resistance  with  wave  length 

The  second  source  of  dissipation  of  power,  that  due  to  ohmic 
resistance,  includes  the  losses  in  the  resistance  of  the  wires,  ground, 
etc.,  of  the  antenna.  Due  to  eddy  currents  and  skin  effect  in 
both  the  wires  and  ground,  this  resistance  will  vary  somewhat 
with  the  wave  length,  being  greater  at  shorter  wave  lengths. 
But  in  an  actual  antenna  these  changes  are  so  small  compared  to 
other  variations  that  we  may  regard  this  component  of  the  total 
antenna  resistance  to  be  almost  constant,  as  it  is  represented 
by  the  straight  line  2  of  Fig.  64.  The  third  source  of  power  dis- 
sipation— i.  e.,  that  due  to  dielectric  absorption — is  a  result  of 
the  fact  that  the  antenna  capacity  is  an  imperfect  condenser. 
The  magnitude  of  this  power  loss  will  depend  upon  the  nature 
and  position  of  imperfect  dielectrics  in  the  field  of  the  antenna. 
Thus  it  has  been  found  that  a  tree  under  an  antenna  may  increase 
the  resistance  of  the  antenna  enormously;  buildings,  wooden 
masts,  and  the  antenna  insulators  also  affect  the  absorption  of 
the  antenna  capacity.  It  is  pointed  out  in  section  34  on  con- 


Radio  Instruments  and  Measurements  83 

densers  that  the  effective  resistance  of  an  absorbing  conden- 
ser is  proportional  to  the  wave  length.  In  the  antenna,  there- 
fore, the  loss  of  power  due  to  dielectric  absorption  may  be  rep- 
resented as  taking  place  in  a  resistance  which  increases  in  propor- 
tion to  the  wave  length.  This  component  of  the  antenna  resist- 
ance is  represented  in  Fig.  64  by  the  straight  line  3. 

The  curve  of  the  total  antenna  resistance  is  obtained  by  com- 
bining these  three  resistance  components  as  in  curve  4  of  the 
same  figure.  This  is  the  typical  resistance  curve  of  an  antenna. 
(See  also  Fig.  91 ,  p.  1 26.)  In  the  case  of  some  antennas  the  resist- 
ance curve  shows  one  or  more  humps  at  certain  wave  lengths. 
This  indicates  the  presence  of  circuits  with  natural  periods  of  oscil- 
lation in  the  vicinity  of  the  antenna,  possibly  the  stays  of  the  an- 
tenna, another  antenna,  or  the  metal  structure  of  a  building;  and 
the  humps  indicate  that,  at  the  particular  wave  length  at  which 
they  occur,  these  extraneous  circuits  are  in  tune  with,  and  are  ab- 
sorbing power  from,  the  antenna.  The  resistance  curve  of  an  an- 
tenna may  be  determined  by  several  of  the  methods  of  resistance 
measurement  given  in  sections  47  to  50  below. 

Measurement  of  Capacity  and  Inductance  of  an  Antenna. — An 
antenna  is  ordinarily  used  with  a  series  loading  coil.  In  the  case 
of  uniform  distribution  of  capacity  and  inductance,  a  formula  and 
table  have  been  given  (pp.  80  and  8 1 )  which  permit  the  wave  length 
of  resonance  to  be  calculated  for  a  given  loading  coil  L  when  the 
quantities  C0  and  L0  are  known.  C0  is  the  low-frequency  capacity 

and  -  !  the  low-frequency  inductance   of  the   antenna.     It  has 

furthermore  been  shown  that  the  resonance  wave  length  can  be 
calculated  with  sufficient  accuracy  from  the  simple  formula  appli- 
cable to  a  circuit  with  lumped  capacity  and  inductance.  The 
capacity  in  the  equivalent  circuit  is  taken  to  be  the  low-frequency 
capacity  C0  of  the  antenna,  and  the  inductance  is  the  sum  of  the 
inductance  of  the  loading  coil  L  and  the  low-frequency  induc- 
tance of  the  antenna  — •  Thus  the  low-frequency  values  of  an- 
tenna capacity  and  inductance  are  sufficient  for  wave-length 
calculations  by  either  formula.  These  low-frequency  values  will 
be  called  simply  the  capacity  Ca  and  inductance  L0  of  the  antenna. 
In  terms  of  the  previous  notation 

C  =C  —1C 

*^8          *^O  —  •'*-'  1 

,    _L0_lLl 

J-^a,  —         — 

3        3 


84  Circular  of  the  Bureau  of  Standards 

and  the  simple  formula  for  the  wave  length  becomes 


"c;  (56) 

where  inductance  is  in  microhenries  and  capacity  in  microfarads. 
Measurement  by  the  Use  of  Two  Loading  Coils. — In  order  to  de- 
termine Ca  and  La  experimentally,  two  loading  coils  of  different 
and  known  values  are  successively  inserted  in  the  antenna 
and  the  wave  lengths  determined  for  which  the  antenna  is  in 
resonance.  This  may  be  done  as  in  Fig.  65,  which  shows  the 
inserted  inductance,  S  a  source  of  oscillations,  and  W  a  stand- 


FlG.   65. — Circuits  for  determining  the  ca- 
pacity and  inductance  of  an  antenna 

ardized  wave  meter.  The  wave  length  of  the  source  is  varied 
until  the  antenna  is  in  resonance  as  indicated  by  the  ammeter  or 
other  indicating  device  on  the  antenna  circuit.  Then  the  antenna 
is  detuned  and  the  wave  length  of  the  source  determined  by  the 
wave  meter.  Two  coils  L  and  L '  are  inserted  and  the  corresponding 
wave  lengths  X  and  X'  are  determined.  Using  the  simple  equation 
(56)  for  lumped  capacity  and  inductance 


Ca 


'  =  i884A/(L'+La)  Ca 


(57) 


Eliminating  Co  between  these  two  equations  and  solving  for  La, 
we  obtain 

r       L>\*-L\»  /,«) 

^        X/2-X2  V5  ' 


Radio  Instruments  and  Measurements  85 

From  the  known  values  of  L,  Lr  ,  X,  and  X'  we  obtain,  therefore, 
the  value  of  La  and  substituting  this  value  in  one  of  the  original 
equations  (preferably  the  one  corresponding  to  the  larger  loading 
coil)  ,  we  obtain  the  value  of  Ca. 

Example,  —  An  illustrative  check  upon  this  method  will  now  be 
given.  Let  us  suppose  that  the  antenna  has  uniformly  distrib- 
uted capacity  and  inductance  of  certain  values  so  that  we  can 
compute  the  wave  lengths  which  would  be  observed  by  experi- 
ment when  the  loading  coils  L  and  L'  are  inserted.  Then,  from 
these  wave  lengths  and  the  values  of  L  and  L'  we  will  compute 
by  the  above  formula  (58)  the  values  of  La  and  Ca,  and  see  how 

closely  they  agree  with  the  original  values  --  and  C0.     We  will 

<u 

take  the  values  C0  =  0.0008  microfarad  and  L0  =  5O  microhenries 
used  before,  and  a  value  for  the  first  loading  coil  of  L  =  100  micro- 
henries, for  which  we  have  previously  found  that  the  wave  length 
would  be  577  meters.  In  addition,  we  will  take  for  the  value  of 
the  second  coil  L'  =400  microhenries.  From  Table  i  we  find  for 

•j-  =  —  —  =  8,  that  co  VCo  Lo  =  -^r  =  0.3465  from  which  we  get  X'  = 

LO       50  A 

27T/ 

—  —  -  =  1088  meters. 


.5 
Substituting  in  formula  (58)  gives 

.       400  (577)2-ioo  (io88)2  .      , 

L'=         (1088)'  -(577)'         =17.3  tn.crohenr.es. 

Using   this   value   of   La   in   equation    (56),   we   have    1088  = 


from  which 

Ca  =  0.000799. 
But  -—  =  16.7  microhenries,  hence  La  differs  from  the  correct 

O 

value  by  3.6  per  cent.  Ca  differs  from  C0  by  only  a  little  over  a 
tenth  of  i  per  cent.  These  values  are  sufficiently  accurate  for 
most  antenna  measurements. 

Corrected  Values.  —  In  measurements  made  with  smaller  values 
for  one  or  both  of  the  inserted  coils,  greater  errors  would 
arise,  but  these  can  be  greatly  reduced  by  a  second  approxima- 
tion. Suppose  that  the  inserted  coils  were  of  values  L  =  50  micro- 
henries, L'  =  200  microhenries.  The  values  of  X  and  X'  would  be 
found  to  be  438.4  and  785.2  meters.  The  value  of  La  would  come 


86  Circular  of  the  Bureau  of  Standards 

out  17.9  microhenries  and  Ca  would  equal  0.000797  microfarad,  La 

differing  from  —  by  7.6  per  cent.     The  errors  are  due  to  the  fact 
o 

that  the  simple  formula  does  not  hold  exactly,  the  per  cent  error 
in  La  being  magnified  on  account  of  the  difference  in  the  squares  of 
the  wave  lengths  in  formula  (58) .  However,  this  approximate  value 
of  La  furnishes  an  approximate  value  of  L0  and  permits  a  close  esti- 
mate of  the  error  in  the  simple  formula.  These  errors  are  given 

in  Table  i  for  different  values  of  j—    As  shown  there,  the  simple 

L0 

formula  leads  to  low  values  for  the  wave  length.  If  the  observed 
wave  lengths  are  corrected  so  as  to  correspond  to  the  simple 
formula,  accurate  values  of  La  and  Ca  will  be  obtained.  Thus  in 
the  case  of  the  5o-microhenry  coil,  since  L0  =  3La  =  53.7,  approx- 
imately, the  ratio  y- =  0.9  and  from  Table  i  the  error  is  0.7  per 

•L"0 

cent. 

Reducing  the  observed  wave  length  X  by  this  amount,  we 
obtain  X  =  435-3  meters.  The  correction  in  the  case  of  the  200- 
microhenry  coil  is  negligible,  hence  X'  is  unchanged. 

Recomputing,  using  X  =  435-3  and  X'  =  785.2,  we  obtain 

La  =  16.6 
and  Ca  =  0.00080 1 

which  are  in  satisfactory  agreement  with  —  and  C0. 

o 

DAMPING 
24.  FREE  OSCILLATIONS 

Up  to  this  point  the  high-frequency  phenomena  considered 
have  been  those  produced  when  current  of  sine- wave  form  is  used. 
Much  of  the  theory  remains  substantially  the  same  for  the  use  of 
so-called  damped  waves.  There  are,  however,  certain  important 
phenomena  which  are  peculiar  to  damped  waves  and  these  will 
be  treated  in  this  section. 

When  a  charged  condenser  is  suddenly  discharged  in  a  circuit 
(Fig.  66)  containing  inductance  and  a  moderate  resistance  in 
series  with  the  capacity,  an  alternating  current  flows,  but  not  of 
sine- wave  form.  If  the  resistance  were  zero,  the  current  would 
have  the  sine-wave  form  and  would  continue  to  flow  forever  with 
undiminished  amplitude.  Every  circuit  contains  resistance,  how- 
ever, which  means  that  the  electric  energy  of  the  current  is  con- 


Radio  Instruments  and  Measurements 


tinually  being  converted  into  heat;  and  also  in  some  circuits  an 
appreciable  amount  of  energy  is  radiated  away  from  the  circuit 
in  electromagnetic  waves.  In  consequence  the  amplitude  of  the 
alternating  current  in  the  circuit  continually  diminishes. 

The  behavior  of  such  a  circuit  is  similar  to  that  of  a  compressed 
spring  to  which  a  weight  is  attached.  When  the  spring  is  re- 
leased a  vibration  commences, 
which  continues  until  the  energy 
of  the  vibration  has  all  been  con- 
sumed by  friction.  If  there  is  no 
external  friction  and  the  spring  is 
very  free  from  internal  strains, 
etc.,  the  vibration  continues  a 
very  long  time;  but  if  the  friction 
is  great,  the  energy  is  all  converted 

.    ,  .    .  . .  T      . ,       FIG.  66. — Oscillatory  circuit  having  resist- 

into  heat  in  a  short  time.     In  the     ^  inductance>  and  capacily  in  series 

same  way,  the  current  in  the  circuit 

under  consideration  is  reduced  very  rapidly  if  the  resistance  is  large. 
A  high-frequency  current  of  continuously  decreasing  amplitude 

is  called  an  "oscillatory"  current,  although  the  term  "oscil- 
latory" is  sometimes  applied  to 
any  current  of  very  high  frequency. 
The  decrease  of  amplitude  is  called 
"damping,"  and  the  current  or 
wave  is  called  a  "damped"  current 
or  wave.  (In  contradistinction  to 
a  damped  wave,  a  wave  in  which 
the  amplitudes  do  not  continuously 
decrease  is  called  a  persistent  or 
sustained  wave  or  oscillation.) 
The  frequency  of  oscillation  in  a 
freely  oscillating  circuit  depends 
only  on  the  inductance  and  capac- 

FIG.  67.— Wave  train  with  a  logarithmic  ity  of  the  circuit,  if  the  resistance 

decrement  of  0.2  js  not  very  large       ^he   damping 

is  determined  by  the  resistance  together  with  the  inductance 
and  capacity.  The  resistance  is  thus  important  in  determining 
the  character  of  the  phenomena.  Damped  currents  are  in  this 
respect  distinct  from  those  of  the  sine-wave  form. 

The  oscillations  which  occur  in  a  simple  circuit  upon  which  no 
external  alternating  emf  is  applied  are  called  the  "free"  oscil- 
lations of  the  circuit.  "Forced"  oscillations,  on  the  other  hand, 


88  Circular  of  the  Bureau  of  Standards 

are  those  impressed  on  the  circuit  by  an  alternating  emf  from  a 
source  outside  the  circuit.  When  free  oscillations  are  produced 
by  the  sudden  discharge  of  a  condenser,  all  of  the  energy  which 
was  stored  in  the  condenser  before  discharge  is  lost  from  the 
circuit  during  the  oscillations.  The  potential  difference  of  the 
condenser,  therefore,  becomes  lower  and  lower  at  every  alternation 
of  the  current.  Since  there  is  no  emf  applied  from  outside  the 
circuit,  the  potential  differences  of  condenser,  resistance,  and 
inductance  must  balance,  and  their  algebraic  sum  be  zero. 

T  di  fidt  /     x 

••Ldt+Rl+c=0  (59) 

This  is  the  same  as  equation  (18)  given  above  in  simple  alternating- 
current  theory,  except  that  e,  the  applied  emf,  is  here  equal  to  o. 
If  the  circuit  contains  a  spark  gap,  the  resistance  R  is  not  a  con- 
,  stant,  and  the  solution  given  immediately  below  does  not  apply. 

Free  oscillations  may  be  produced  by  another  method  than  the 
sudden  discharge  of  the  condenser  in  the  circuit.  If  current  is 
produced  in  the  circuit  by  induction  and  the  inducing  emf  is 
suddenly  cut  off,  free  oscillations  are  produced.  The  quenched 
gap  is  the  means  utilized  in  practice  for  suddenly  cutting  off  the 
inducing  emf,  the  gap  being  in  a  circuit  closely  coupled  to  the 
oscillating  circuit. 

The  solution  of  equation  (59)  for  any  circuit  in  which  the 
resistance  is  not  extremely  great,  is 

i  =  I0e~at  sin  u>t 

where  /0  is  the  initial  current  amplitude,  a  is  the  damping  factor, 
and  (a  is  2ir  times  frequency  of  oscillation.  The  values  of  these 
constants  are,  V0  being  the  initial  potential  difference  across  the 
condenser, 

/o  = 
R 

<X=—f 
2L 


03    = 


This  last  expression  is  an  approximation  for 


Radio  Instruments  and  Measurements  89 

The  resistance  of  practically  all  high-frequency  circuits  is  so 
small  that  the  approximation  is  sufficiently  accurate.  Using 
these  relations,  it  follows  that 


_ 

*L 


sin 


The  frequency  of  oscillation  is  called  the  "natural  frequency" 
of  the  circuit.  It  should  be  noted  that  the  natural  frequency  of 
a  circuit  in  which  the  resistance  is  not  very  large  is  the  same  as 
the  frequency  of  resonance  to  an  electromotive  force  impressed 
upon  the  circuit.  Thus  it  was  shown  in  section  n  above,  that 
when  an  alternating  emf  is  impressed  upon  an  inductance  and 

capacity  in  series  resonance  occurs  when  coL  =  — -.     This  relation 

is  the  same  as  co=    /— ,  the  expression  above   for  the  natural 

frequency. 

Effective  Value  of  Damped  Current. — In  the  various  spark  types 
of  radio  transmitters  the  condenser  is  charged  at  regular  intervals, 
each  charge  being  followed  by  an  oscillatory  discharge.  Fig.  68 


FiG.  68. — Series  of  wave  trains  similar  to  that  generated  by  a  damped  wave  source 

roughly  indicates  the  successive  discharges;  actual  current  curves 
are  more  complicated  than  this.  The  energy  in  each  train  of 
oscillations  is  practically  all  dissipated  before  the  next  train  begins. 
Under  these  conditions  the  effective  (root-mean-square)  value  of 
the  current  /  in  terms  of  the  initial  maximum  current  I0  is  easily 
found,  as  follows  (7=current  which  would  be  indicated  by  the 
steady  deflection  of  an  ammeter) : 

Let  N  =  number  of  trains  of  oscillations  per  second. 

Wt  =  energy  dissipated  per  train  of  waves. 

W3  =  magnetic  energy  associated  with  current  at  beginning 
of  each  train. 

Then  W 


90  Circular  of  the  Bureau  of  Standards 

These  two  energies  must  be  equal,  since  it  is  assumed  that  the 
energy  in  each  train  of  oscillations  is  dissipated  before  the  next 
train  begins. 

RP_ 

N 

P  =   *R 
Since 

R 

7,      N 

P  = /o  (60) 

40; 

25.  LOGARITHMIC  DECREMENT 

The  rate  of  decrease  of  the  current  amplitudes  during  a  train 
of  waves  is  shown  by  the  damping  factor  a  defined  above. 
When  this  is  large,  the  curve  ABC,  drawn  through  the  maxima 
of  the  oscillations  in  Fig.  67,  approaches  the  axis  quickly.  The 
shape  of  the  train  of  waves  is  such  that  the  ratio  of  any  maximum 
to  the  next  one  following  is  a  constant.  The  rate  of  decrease  of 
the  amplitudes  in  a  wave  train  is  also  indicated  by  the  logarithmic 
decrement,  which  is  defined  to  be  the  natural  logarithm  of  the 
ratio  of  two  successive  maxima  in  the  same  direction.6  The  rela- 
tion between  the  damping  factor  and  the  logarithmic  decrement 
may  be  found  as  follows:  The  ratio  of  one  maximum  of  current 
to  the  following  is 


where  T  is  the  period  of  oscillation  and  equals  —     This  ratio  = 


lira 
aT 


ea   =  e  "  .    The  napierian  logarithm  of  the  ratio  is  the  logarith- 
mic decrement  5  which  consequently  equals  -  —     Since  a  =  -j- 

x         R 

0  =7T  —  j- 


The  logarithmic  decrement  is  thus  equal   to  TT  times   the  ratio 
between  the  resistance  and  the  inductive  reactance.7     It  is  also 

8  A  few  writers,  Fleming  in  particular,  define  the  logarithmic  decrement  as  the  natural  logarithm  of  the 
ratio  of  two  successive  maxima  in  opposite  directions.  The  Fleming  decrement  is  thus  equal  to  one-half 
the  decrement  as  defined  above. 

7  Reactance  is  ordinarily  calculated  from  the  simple  expression  for  u>,  which  is  not  rigorously  true  when 
the  decrement  is  very  large. 


Radio  Instruments  and  Measurements  91 

equal  to  TT  times  the  ratio  between  the  resistance  and  capacitive 
reactance,7  for 

co  =  — p=  or  coL  =  —  and  therefore 
VLC  coC 


Thus  the  logarithmic  decrement  is  TT  times  the  reciprocal  of  the 
sharpness  of  resonance  (discussed  above  on  p.  36). 

The  logarithmic  decrement  expressed  in  terms  of  the  three 
quantities  resistance,  inductance,  and  capacity  is 8 


Interpretations  of  Logarithmic  Decrement.  —  The  decrement  has 
been  given  in  terms  of  (i)  a  ratio  of  current  amplitudes,  (2)  a 
ratio  of  impedance  components,  and  (3)  the  reciprocal  of  the 
quantity  called  sharpness  of  resonance.  A  fourth  interesting 
interpretation  is  in  terms  of  an  energy  ratio.  It  is  readily  shown 
that  8  =  ^2  the  ratio  of  the  average  energy  dissipated  per  cycle 
to  the  average  magnetic  energy  at  the  current  maxima,  as  follows: 

RP 
-^-JL-   JL 


where  /  is  the  effective  current  as  measured  by  an  ammeter. 

-j-  =  average  energy  dissipated  per  cycle,  since  RP  =  average 

energy  dissipated  per  second. 

LP  =  average  (L/k2)  (where  7k2  =  average  current  square  during 
the  k  'th  cycle)  ,  since  P  =  average  /k2, 

(I      2\ 
L—    -  J,  (where  /max  =  maximum  current  during  the  fc'th 

cycle),  since  7k2  =  /4/max2,  just  as  in  the  case  of  undamped  currents. 

/     I      2\ 
But  (  L    max  j  is  the  magnetic  energy  at  the  current  maximum 

during  the  k  'th  cycle.  Therefore  LP  =  average  magnetic  energy 
at  the  current  maxima.  Hence  the  decrement  is  one-naif  the  ratio 
of  the  average  energy  dissipated  per  cycle  to  the  average  magnetic 
energy  at  the  current  maxima.  This  is  true  when  the  energy  is 
lost  from  the  circuit  by  radiation  as  well  as  when  lost  by  heating. 

7  Reactance  is  ordinarily  calculated  from  the  simple  expression  for  w,  which  is  not  rigorously  true  when 
the  decrement  is  very  large. 

8  This  expression  for  the  decrement  does  not  hold  when  the  decrement  is  extremely  large. 


92  Circular  of  the  Bureau  of  Standards 

Number  of  Oscillations  in  a  Wave  Train.  —  There  are  theoret- 
ically an  infinite  number  of  oscillations  in  a  wave  train.  In  prac- 
tice, however,  the  wave  train  may  be  considered  ended  when  the 
oscillations  are  reduced  to  a  negligible  amplitude.  The  fraction 
of  the  initial  amplitude  that  is  considered  negligible  depends  on 
the  use  to  which  the  oscillatory  current  is  put.  For  a  given  ratio  of 

initial  amplitude  to  final  amplitude,  -p,  the  number  of  complete 

*n 

oscillations  n  is  given  by 

/o 

log*     - 


and  the  number  of  maxima  or  of  semioscillations 

/o 
2    lo 


5 
The  oscillations  after  the  amplitude  is  reduced  to  o.oi  of  its  initial 

value  can  usually  be   considered  negligible.     For  -2  =  100,  the 

*  n 

number  of  oscillations  =  4—.     For  example,  the  number  of  oscil- 

5 

lations  in  which  a  current  having  5  =  0.2  falls  off  to  i  per  cent  of 
its  initial  value  is  equal  to  23. 

26.  PRINCIPLES  OF  DECREMENT  MEASUREMENT 

A  number  of  so-called  methods  of  measuring  decrement  are  in 
reality  measurements  of    resistance.     From  the    resistance  the 

D     [7* 

logarithmic  decrement  of  a  circuit  is  calculated  by  5==  —  j=r-  or 


one  of  the  related  formulas.  Any  method  for  measuring  the  re- 
sistance of  a  circuit  thus  enables  one  to  calculate  the  decrement. 
The  value  so  obtained  is  the  decrement  of  the  current  that  would 
flow  in  the  circuit  if  free  oscillations  were  suddenly  started,  but 
not  in  general  the  decrement  of  the  current  used  in  making  the 
measurement.  The  methods  available  for  such  resistance  meas- 
urements are  summarized  below  in  sections  47  to  50.  Only  those 
methods  in  which  damped  oscillations  are  used  can  be  considered 
actual  measurements  of  decrement. 

.There  are  two  classes  of  genuine  decrement  measurement,  one 
in  which  free  oscillations  are  used  and  one  in  which  a  damped 
electromotive  force  is  impressed  on  the  circuit  so  that  both  free 
and  forced  oscillations  contribute  to  the  current.  Free  oscilla- 


Radio  Instruments  and  Measurements 


93 


tions  are  obtained  in  the  case  of  pure  impulse  excitation.  It  is 
very  difficult  in  practice  to  obtain  such  excitation.  Assuming, 
however,  that  free  oscillations  are  produced  by  the  sudden  dis- 
charge of  the  condenser  at  a  constant  potential  difference  N  times 
per  second,  the  effective  current  is  given  by  equation  (60), 


Since 


4" 


co5 


8>  p 


This  may  be  written,  725  =  constant.  Let  /  be  the  current  for  a 
certain  resistance  in  the  circuit,  and  Iv  be  the  current  when  a 
resistance  R^  is  added  so  as  to  increase  the  decrement  by  an 
amount  8t. 

725=712(5  +  51) 

•••5  =  5iFZ71  (61) 

Bjerknes  Methods. — A  method  of  measuring  the  high-frequency 
resistance  of  a  circuit  using  undamped  currents  has  been  described 
on  page  38  above  and  others  are  described  in  sections  49  and  50 

R 


FIG.  69. — Inductively  coupled  circuits  for 
decrement  measurements 

below.  These  methods  have  been  extended  to  the  measurement 
of  resistance  and  decrement  by  the  use  of  damped  waves.  When 
a  damped  emf  is  impressed  on  the  circuit  both  free  and  forced 
oscillations  exist,  and  the  measurement  is  an  actual  measurement 
of  decrement.  For  a  circuit  77  (Fig.  69)  very  loosely  coupled  to 
a  circuit  7  in  which  oscillations  are  generated,  Bjerknes  9  showed 
that  when  the  two  circuits  are  in  resonance, 

JVEo2 


i6L2a'a(a' 


9  See  reference  No.  42,  Appendix  2. 


94  Circular  of  the  Bureau  of  Standards 

where  E0  is  the  maximum  value  of  the  impressed  electromotive 
force,  N  the  number  of  trains  of  waves  per  second,  a.'  the  damping 
factor  of  the  emf  due  to  circuit  /  impressed  on  circuit  II,  and 
a  the  damping  factor  of  the  second  circuit.  The  circuit  /  may  be 
a  great  distance  from  II,  and  may  even  be  a  distant  radiating 
antenna.  The  equation  holds  only  when  a'  and  a  are  small  in 
comparison  with  <o,  or  when  the  decrements  6'  and  5  are  small 
compared  with  2?r. 

In  the  resistance  variation  method  of  determining  decrement, 
the  resistance  of  the  second  circuit  is  increased  by  an  amount  R: 
changing  a.  to  a-}-a^  and  the  original  current  7  to  some  other 
value  /.  We  then  have 


T2_ 
*•!• 


•'     8' 


Oi 

Since  —  =  -T-I  it  follows  that 
a      0 


1 6L2a'  (a  +  aj  (a'  +  a  +  aj 
that 

P     (8  +  50  (8' 
,2  8(5'  +  8) 


(62) 


This  can  be  solved  either  for  8'  if  5  is  known  or  for  6  if  8'  is  known; 
5'  being  the  decrement  of  the  wave  emitted  by  circuit  I  and  im- 
pressed on  circuit  //,  8  being  the  decrement  of  circuit  //,  and  St 
the  amount  by  which  8  is  increased  by  adding  resistance.  The 
solution  for  8'  is 

72  _  /  2 
2581  +  512-  —r~y 

'  (63) 


•      ~M      S        SJ 

r  2       d-di 
Ji 

This  may  be  simplified  by  choosing  the  resistance  inserted  such 
that  81=S;  then 

2  —  J2 

i 


=  5 

° 


_ 

/2-2/12 


Another  convenient  simplified  procedure  is  to  vary  the  inserted 
resistance  until  the  square  of  the  current  is  reduced  to  one-half  its 

I2  —  7  2 
previous  value,  then  —  p~  =  i  ,  and 


(65) 


Radio  Instruments  and  Measurements  95 

When  5  is  the  unknown,  the  direct  solution  for  5  in  terms  of  5' 
is  complicated;  equation  (62)  is  in  fact  more  convenient  in  this 
case  than  an  explicit  solution.  A  useful  form  of  the  solution  for  5 
in  certain  cases  is 


s 

where  K  =  i  +  ^7-7- 


This  is  discussed  on  page  184  below.  It  should  not  be  forgotten 
that  these  formulas  apply  only  when  the  coupling  is  very  loose 
and  both  decrements  are  small. 

The  reactance  variation  method  of  measuring  decrement  is 
similar  in  procedure  to  the  resistance  variation  method,  two 
observations  of  current  being  taken  with  different  reactances. 
The  method  is  described  on  pages  186  to  199  below,  formulas  (96) 
to  (100)  showing  the  method  of  calculation.  The  measurement 
of  the  decrement  of  a  wave  is  treated  in  sections  54  and  55. 

35601°— 18 7 


PART  H.—  INSTRUMENTS  AND  METHODS  OF 
HIGH-FREQUENCY  MEASUREMENT 


27.  GENERAL  PRINCIPLES 

There  is  considerable  difficulty  in  attaining  high  accuracy  in 
measurements  at  radio  frequencies.  Much  of  this  is  due  to  the 
fact  that  the  quantities  to  be  measured  or  upon  which  the  meas- 
urement depends  are  generally  small  and  sometimes  not  definitely 
localized  in  the  circuits.  Thus  the  inductances  and  capacities 
used  in  the  measuring  circuits  are  so  small  that  the  effect  upon 
these  quantities  of  lead  wires,  indicating  instruments,  surround- 
ings, etc.,  must  be  carefully  considered.  The  capacity  of  the  in- 
ductance coil  and  sometimes  even  the  inductance  within  the  con- 
denser are  of  importance.  In  order  to  minimize  these  various 
effects  it  is  generally  best  to  use  measuring  circuits  and  methods 
which  are  the  least  complicated.  On  this  account  simple  circuits 
and  substitution  methods  in  which  the  determination  depends 
upon  deflections  are  usually  used  in  preference  to  more  com- 
plicated methods. 

In  addition  to  the  uncertainty  or  the  distributed  character  of 
some  of  the  quantities  to  be  measured,  there  are  other  limitations 
upon  the  accuracy  of  radio  measurements.  The  usual  ones  are 
the  variation  with  frequency  of  current  distribution,  inductance, 
resistance,  etc.,  and  the  difficulty  of  supplying  high-frequency 
current  of  sufficient  constancy.  The  latter  limitation  is  entirely 
overcome  by  the  use  of  the  electron  tube  as  a  source  of  current 
but  is  troublesome  when  a  buzzer,  spark,  or  arc  is  used.  As  to 
the  other  difficulty,  the  variations  of  inductance,  etc.,  with  fre- 
quency, while  these  variations  have  a  profound  effect,  they  are 
generally  subject  to  control.  They  must  be  reckoned  with  in  the 
design  of  high-frequency  ammeters  and  other  instruments,  but 
the  quantities  have  definite  values  at  a  particular  frequency  under 
definite  conditions,  and  their  effect  can  usually  be  determined  by 
calculation  or  measurement. 

It  is  not  always  possible  to  determine  the  effects  of  the  capacities 
of  accessory  apparatus  and  surroundings,  nor  to  eliminate  them, 
96 


Radio  Instruments  and  Measurements  97 

and  thus  they  remain  the  principal  limitation  upon  the  accuracy 
of  measurements.  These  stray  capacities  include  the  capacities 
of  leads,  instrument  cases,  table  tops,  walls,  and  the  observer. 
They  may  not  only  be  indeterminate  but  may  vary  in  an  irregular 
manner.  Means  of  dealing  with  them  are  discussed  in  section  31 
below. 

Relative  Importance  of  Measurements. — On  account  of  the 
requirement  of  simplicity  in  radio  measurements,  the  methods 
available  are  quite  different,  and  are  fewer  in  number,  than  in  the 
case  of  low-frequency  or  direct-current  measurements.  The  wave 
meter  and  the  ammeter  are  the  principal  measuring  instruments 
used,  and  with  these  two  most  of  the  important  radio  measure- 
ments can  be  made.  The  principal  additional  pieces  of  apparatus 
are  condensers  and  coils.  These  are  the  essential  constituents  of 
radio  circuits.  It  may  be  recalled  that  the  principal  difference 
between  the  phenomena  of  high  and  low  frequency  is  the  impor- 
tance of  inductance  and  capacity  at  high  frequency  as  compared 
with  the  predominance  of  resistance  in  low -frequency  phenomena. 
Thus  the  resistance  of  circuits  is  the  chief  factor  determining  cur- 
rent flow  in  the  case  of  direct  current,  and  this  is  largely  true  in 
low-frequency  alternating  current;  in  high-frequency  circuits,  on 
the  contrary,  resistance  is  of  minor  importance,  and  the  flow  of 
current  is  mainly  determined  by  reactance,  a  quantity  dependent 
upon  inductance  and  capacity.  The  significance  of  the  role 
played  by  reactance  is  shown  by  sections  13-18  above,  in  which 
many  problems  of  radio  circuits  are  worked  out  by  the  use  of 
reactance  without  any  consideration  of  resistance. 

The  increase  of  resistance  with  frequency  renders  all  the  more 
striking  the  general  fact  that  resistance  is  of  less  importance  in 
determining  the  distribution  of  high-frequency  currents  than 
capacity  and  inductance.  While  resistance  is  not  of  primary 
importance  in  radio  circuits,  nevertheless  it  must  in  many  cases 
be  known  accurately,  and  on  account  of  the  change  with  fre- 
quency must  be  measured  at  the  particular  frequency  for  which  its 
value  is  needed. 

WAVE  METERS 

28.  THE  FUNDAMENTAL  RADIO  INSTRUMENT 

The  most  generally  useful  measuring  instrument  at  radio 
frequencies  is  the  wave  meter.  It  measures  primarily  frequency, 
which  is  customarily  expressed  in  terms  of  wave  length.  It  may 


98  Circular  of  the  Bureau  of  Standards 

also  be  used,  however,  to  compare  inductances  and  capacities,  to 
measure  resistance  and  decrement,  to  determine  resonance  curves, 
and,  in  fact,  to  make  most  of  the  measurements  required  for  radio 
work.  When  equipped  with  a  buzzer  or  other  source  of  power,  a 
wave  meter  may  also  be  used  as  a  generator  of  currents  of  known 
frequency.  Wave  meters  may  in  general  be  divided  into  two 
classes,  standard  and  commercial.  The  former  type  is  so  designed 
as  to  permit  its  calibration  to  be  derived  from  a  knowledge  of  the 
value  of  its  constituent  elements;  the  latter  is  designed  for  com- 
pactness and  convenience,  and  is  ordinarily  calibrated  by  com- 
parison with  a  standard. 

In  its  usual  form  a  wave  meter  is  essentially  a  simple  radio 
circuit,  consisting  of  an  inductance  coil  and  condenser  in  series, 
with  an  ammeter  or  other  device  to  indicate  either  the  current 
flowing  in  the  circuit  or  the  voltage  across  the  whole  or  a  part  of 
the  capacity  or  inductance.  Either  the  inductance  or  capacity  is 
made  variable  and  sometimes  both.  Usually  the  condenser  is 
variable,  and  a  number  of  inductance  coils  of  different  values  are 
provided.  Each  coil  in  connection  with  the  condenser  gives  a 
definite  range  of  wave  lengths,  and  the  different  coils  are  so  chosen 
that  these  separate  ranges  overlap  so  as  to  cover  the  complete 
range  of  wave  lengths  desired. 

To  measure  the  wave  length  of  the  oscillations  given  out  by  a 
source  the  wave  meter  is  loosely  coupled  to  it  and  the  variable 
condenser  adjusted  to  that  setting  which  gives  the  maximum 
current  in  the  indicating  device.  In  this  condition  of  resonance 
the  frequency  is  given  by  equation  (24),  page  32,  and  the  wave 
length  is  related  to  the  inductance  and  capacity  in  the  circuit 
according  to  the  expression 

^  =  k^/LC  (67) 

in  which  k  is  a  constant,  values  of  which  are  given  on  page  312  for 
L  and  C  in  various  units.  In  a  standard  wave  meter  the  values  L 
and  C  are  known  and  the  wave  length  may  be  computed.  In  the 
commercial  type  the  values  of  the  wave  length  are  determined  for 
each  coil  and  each  condenser  setting  by  comparison  with  a  standard 
wave  meter,  and  may  be  read  off  from  curves  or  from  the  scale  of 
the  variable  condenser.  This  scale  may  be  graduated  in  degrees, 
arbitrary  divisions,  wave  length,  or  even  in  terms  of  frequency  or 
of  capacity.  Sometimes  several  scales  are  put  on  one  instrument. 


Radio  Instruments  and  Measurements  99 

29.  CALIBRATION  OF  A  STANDARD  WAVE  METER 

The  most  direct  method  for  the  wave  length  calibration  of  a 
standard  or  commercial  wave  meter  is  a  comparison  with  a  high- 
frequency  alternator.  From  the  speed  of  the  machine  and  the 
number  of  poles  or  other  structural  data  the  frequency  of  alterna- 
tion can  be  computed  directly.  The  range  of  such  alternators  is, 
however,  limited;  the  usual  construction  does  not  furnish  a  wave 
length  shorter  than  3000  meters. 

Low-Frequency  Measurements  as  Basis. — Another  method  of  pro- 
cedure consists  in  measuring  separately  the  inductance  and  capac- 
ity of  the  standard  wave  meter  and  computing  the  wave  length 
from  these  quantities  when  combined.  The  capacity  is  measured 
at  low  frequencies  and  assumed  to  be  the  same  at  high  frequencies. 
This  assumption  is  justifiable  if  the  condenser  is  one  that  is  free 
from  dielectric  absorption  and  has  an  insulation  resistance  so  high 
as  not  to  affect  the  low-frequency  capacity.  A  convenient  means 
of  measuring  capacity  with  low-frequency  interrupted  direct  cur- 
rent is  the  Maxwell  bridge.  This  method  10  permits  the  determina- 
tion of  capacity  in  terms  of  resistance  and  the  frequency  of  the 
commutator,  tuning  fork,  or  other  device  which  charges  and  dis- 
charges the  condenser.  Capacities  can  be  determined  by  this 
method  to  an  accuracy  of  i  micromicrofarad,  which  is  sufficient 
for  radio  measurements. 

The  capacity  of  an  unshielded  condenser  will  depend  upon  its 
position  in  the  circuit  and  its  proximity  to  the  body  of  the  observer, 
walls  of  the  room,  etc.  The  condenser  should  therefore  have  one 
set  of  its  plates  connected  to  a  metal  shield,  and  the  shield  con- 
nected to  ground,  both  when  it  is  being  calibrated  at  low  fre- 
quency and  when  it  is  being  used  in  the  wave-meter  circuit.  The 
ground  connection  to  a  radio  circuit  should  be  a  thick  conductor, 
run  by  as  short  a  path  as  possible  to  ground. 

The  inductance  coils,  if  wound  with  properly  stranded  wire  so 
that  there  is  little  skin  effect  and  if  free  from  other  metal  so  that 
there  is  no  eddy-current  effect,  may  have  their  inductances  com- 
puted or  measured  at  low  frequency  and  the  values  used  at  high 
frequencies;  provided,  however,  that  condensers  of  such  large 
capacity  values  are  used  with  the  coils  that  the  capacities  of  the 
coils  themselves  are  negligible.  In  certain  coils  of  small  radius  and 
wound  with  heavy  stranded  conductor  with  inferior  insulation 

10  See  reference  No.  174,  Appendix  2. 


TOO 


Circular  of  the  Bureau  of  Standards 


between  strands  it  was  found  that  at  radio  frequencies  the  in- 
ductance was  reduced  by  more  than  i  per  cent. 

Use  of  Calculable  Inductance  Standard. — A  good  method  of 
determining  the  high-frequency  inductances  of  the  standard  coils 
is  to  compare  at  high  frequencies  the  smallest  coil  of  the  wave- 
meter  with  an  inductance  of  simple  form  such  as  a  large  single- 
turn  square  or  rectangle,  the  value  of  which  can  be  computed. 
(Use  formulas  in  sec.  68.)  The  larger  coils  are  then  compared 
with  the  small  coil  as  outlined  below.  If  the  square  is  made  of 
round  wire  the  inductance  may  be  calculated  very  accurately  by 
(137).  The  following  corrections  must  be  considered.  (See 
Fig.  70.) 

First,  the  condenser  calibration  takes  account  of  the  capacity 
only  from  the  terminals  A,  B,  the  binding  posts  of  the  condenser. 

In  addition  to  this  there  is  the 
capacity  between  the  leads  to  the 
square  and  between  the  leads, 
square,  etc.,  and  the  shield  or  case 
of  the  condenser.  This  correc- 
tion may  be  determined  experi- 
mentally by  two  methods.  In 
one  the  square  is  cut,  say  at  the 
points  D  and  E,  and  some  other 
coil  connected  to  the  terminals 
A,  B,  and  coupled  to  a  source  of 
oscillations.  Keeping  the  wave 
length  of  the  source  constant  the  setting  of  the  condenser  C  for 
resonance  is  obtained  with  the  terminals  of  the  square  connected 
to  the  condenser  terminals  and  then  with  them  disconnected. 
The  capacity  of  the  condenser  for  resonance  in  the  latter  case  will 
be  greater  than  in  the  former  by  an  amount  very  close  to  the 
required  correction. 

Capacity  Correction  Determined  by  Pliotron  Harmonics. — In  the 
other  method  the  harmonics  emitted  by  such  a  source  as  the  plio- 
tron  generator  are  used.  In  the  current  furnished  by  such  a  gene- 
rator all  of  the  harmonics  are  present.  The  circuit  is  first  tuned 
to  the  fundamental  by  means  of  the  condenser  C  and  then  to  the 
harmonic,  say  of  double  frequency.  Since  the  inductance  is  kept 
constant  and  the  frequency  varies  inversely  as  the  square  root  of 
the  capacity,  the  total  capacity  required  to  tune  for  the  har- 
monic will  be  one-quarter  of  that  for  the  fundamental.  The 


FIG.  70. — Circuit  consisting  of  a  calcula- 
ble inductance  standard  and  a  shielded 
standard  condenser 


Radio  Instruments  and  Measurements  101 

method  depends  on  the  assumption  that  the  frequencies  of  the 
harmonics  are  exact  integral  multiples  of  the  fundamental  fre- 
quency. Letting  Cf  and  C2f  represent  the  capacities  of  the  con- 
denser for  the  fundamental  and  harmonic,  and  c  the  extraneous 
capacity,  we  have 


therefore 


Determination  of  Inductance  Corrections.  —  The  second  correc- 
tion which  must  be  considered  involves  the  inductance  of  the  leads 
connecting  the  square  and  the  condenser,  the  inductance  of  the 
leads  within  the  condenser,  and  the  inductance  of  the  indicating 
instrument.  The  first  of  these  inductances  can  be  calculated 
with  sufficient  accuracy,  making  use  of  the  formulas  for  straight 
wires  and  two  parallel  wires  given  on  page  245.  The  inductance 
of  the  leads  within  the  condenser  is  very  small  in  a  well-designed 
condenser.  It  can  be  determined  by  comparing  two  squares  of 
different  inductances,  one  of  which  should  be  small.  If  Lt  and 
L2  are  the  inductances  of  the  squares  and  leads,  /  is  the  inductance 
of  the  condenser  and  Ct  and  C2  are  the  values  of  capacity  at  which 
resonance  is  obtained  with  the  two  squares  at  a  given  wave 
length,  then 


and 

T    r  —  T  r 

-          J-^-t     \ i         •^^'2    2 

Ca-Ci 

If  the  indicating  instrument  is  inserted  directly  in  the  circuit,  its 
inductance,  if  appreciable,  must  be  added  to  that  of  the  square. 
A  sensitive  hot-wire  ammeter  may  have  0.5  microhenry  induc- 
tance, which  may  be  a  large  fraction  of  the  inductance  of  the 
circuit.  The  value  can  be  readily  determined  by  inserting  the 
ammeter  in  a  circuit  previously  tuned  to  resonance  by  means  of 
another  indicating  instrument  and  noting  the  change  in  the  con- 
denser setting  required  to  xretune  the  circuit.  A  crossed-wire 
thermoelement,  if  made  with  very  short  heating  wires  and  leads, 
has  a  very  small  inductance  and  may  usually  be  inserted  directly 
in  the  circuit  without  appreciable  error.  Instead  of  inserting 
the  indicating  instrument  directly  in  the  circuit,  it  is  possible  to 
connect  one  or  more  turns  of  wire  to  the  instrument  and  couple 


IO2  Circular  of  the  Bureau  of  Standards 

it  with  the  circuit.  If  sufficient  power  is  available,  the  coupling 
may  be  made  so  loose  that  the  indicating  instrument  has  no 
appreciable  effect  upon  the  inductance  of  the  circuit. 

Comparison  of  Inductance  Standards. — The  method  of  comparison 
of  the  standard  square  and  a  standard  coil  is  shown  in  Fig.  71. 
/  is  the  source,  //  the  circuit  of  the  square  and  condenser,  and  777 
the  standard  coil  and  condenser.  The  source  /  is  set  at  a  given 
wave  length  and  the  condenser  of  II  is  tuned  to  resonance,  and  the 
condenser  reading  obtained.  Then  II  is  detuned  and  777  is  tuned 
and  the  condenser  read.  Both  the  inductance  and  capacity  are 
known  for  the  circuit  II,  and  since  777  is  tuned  to  the  same  wave 
length  the  product  of  capacity  and  inductance  for  this  circuit  must 


FIG.  71. — Circuits  involved  in  the  comparison 
of  inductance  standards 

be  the  same  as  that  for  II.  Dividing  this  product  by  the  observed 
capacity  for  ///  gives  the  apparent  inductance  in  777.  This  proc- 
ess is  repeated  over  a  wide  range  of  wave  lengths  and  the  appar- 
ent inductance  of  the  coil  is  found  to  increase  as  the  wave  length 
is  reduced.  It  is  shown  above  (p.  64)  that  this  is  a  result  of  the 
fact  that  the  capacity  of  the  coil  has  been  included,  together  with 
its  inductance;  hence  the  term  "apparent"  inductance  is  used. 
It  is  shown  in  section  19  that,  instead  of  using  the  values  of  the 
apparent  inductance  for  each  wave  length,  it  is  possible  to  rep- 
resent the  coil  very  accurately  by  a  fixed  value  for  the  inductance 
of  the  coil  called  the  ' '  pure ' '  inductance  and  a  value  for  the  coil 
capacity.  When  the  coil  is  used  with  a  condenser  to  form  a 
.wave-meter  circuit,  the  capacity  of  the  coil  is  added  to  that  of 


Radio  Instruments  and  Measurements  103 

the  condenser  and  the  wave  length  computed  from  this  total 
capacity  and  the  pure  inductance.  In  the  absence  of  skin  effect, 
etc.,  the  pure  inductance  of  the  coil  is  identical  with  its  low- 
frequency  inductance.  Having  determined  the  pure  inductance 
and  capacity  of  the  smallest  coil  of  the  standard  wave  meter,  this 
coil  may  be  used  to  determine  the  values  of  the  other  coils  by 
the  same  comparison  method.  The  small  coil  with  a  large  con- 
denser is  tuned  to  the  wave  length  of  the  source,  and  a  larger 
coil  with  smaller  condenser  is  then  tuned  to  the  same  wave  length. 
The  product  L1Cl  for  the  small  coil,  which  is  known,  is  equal  to 
the  product  L2C2  for  the  larger  coil  from  which  the  apparent 
inductance  of  the  larger  coil 


*<a-    c 

L-2 

The  apparent  inductance  of  all  the  larger  coils  may  thus  be  deter- 
mined by  stepping  from  coil  to  coil. 

Coil  Measurements  by  Pliotron  Harmonics. — Some  errors  will 
arise  in  these  successive  comparisons  which  makes  it  desirable  to 
be  able  to  compare  directly  the  large  coils  with  the  small.  By 
making  use  of  the  harmonics  of  a  generator  of  the  pliotron  type 
it  is  possible  to  compare  directly  small  and  large  coils  without  the 
requirement  of  very  large  condensers,  and  the  capacity  of  each 
coil  can  be  readily  determined.  In  comparing  the  coils,  a  cir- 
cuit containing  the  large  coil  is  tuned  to  the  fundamental.  The 
circuit  with  the  small  coil  is  then  tuned  to  the  harmonic  of  fre- 
quency two,  three,  or  more  times  the  fundamental.  The  wave 
length  in  the  latter  case  will  be  one-half,  one- third,  etc.,  of  that 
in  the  former;  the  product  of  L  and  C  will  be  one-fourth,  one- 
ninth,  etc.  Thus,  if  C2  is  the  known  capacity,  L2  the  unknown 
inductance  for  the  large  coil,  and  L^  and  C\  are  the  known  values 
for  the  small  coil  and  the  harmonic  of  threefold  frequency  is 
used,  we  have 

T     /"*  7"     /"* 

L      Q^. 

-^-'2  —  j       f~* 

L2  is  the  apparent  inductance.  The  capacity  of  the  coil  is  then 
obtained  in  a  manner  similar  to  that  given  above  as  one  of  the 
methods  of  getting  the  capacity  of  leads,  etc.,  of  the  square.  The 
coil  plus  condenser  is  tuned  to  the  fundamental  and  then  to  the 
harmonic  of  double  frequency.  The  condenser  capacity  will.be 


IO4  Circular  of  the  Bureau  of  Standards 

roughly  one-quarter  in  the  latter  case  and  should  be  quite  small. 

x-t  j~* 

As    above,  we  obtain  for  the  coil  capacity    c  =  —       — .     It  is 

O 

advisable  to  repeat  the  measurement  at  several  differing  funda- 
mental frequencies  and  average  the  values  of  c  obtained.  The 
pure  inductance  Lp  is  then  determined,  making  use  of  known 
values  of  L2C2,  as  determined  above,  for 


.       L->iL,  ^ 

L  =L  * 

The  mean  based  on  several  values  should  be  obtained. 

In  the  above  comparisons  the  leads  of  the  coils  have  been  con- 
sidered as  a  part  of  the  coil,  contributing  both  to  the  inductance 
and  capacity  of  the  whole.  These  leads  should  be  fixed  and 
definite  and  should  be  of  sufficient  length  and  otherwise  designed 
so  that  the  coil  constants  will  not  be  appreciably  altered  on 
account  of  eddy  currents  in  the  condenser  case  or  capacity  to  it, 
when  the  coil  is  connected  to  the  condenser  to  form  a  wave  meter. 

30.  STANDARDIZATION  OF  A  COMMERCIAL  WAVE  METER 

A  commercial  wave  meter  is  generally  equipped  with  one  or 
more  indicating  devices,  of  which  the  hot-wire  ammeter  and 
crystal  detector  with  phones  are  especially  important.  In 
addition,  it  is  customary  to  provide  a  buzzer  circuit,  so  that 
oscillations  of  known  wave  length  may  be  generated  by  the 
wave  meter.  The  indications  of  the  wave  meter  will  be  some- 
what different,  depending  upon  the  way  it  is  operated — that  is, 
whether  one  or  the  other  of  the  indicating  devices  is  used  or 
whether  it  is  used  as  a  source  of  oscillations.  Hence,  in  each  of 
these  cases  it  will  usually  be  necessary  to  have  a  separate  cali- 
bration. Calibration  of  wave  meter  used  as  a  source  is  treated  on 
page  1 08  below. 

Wave  Meter  with  Ammeter. — When  the  hot-wire  ammeter  is  in 
use,  it  is  either  inserted  in  the  circuit  directly,  when,  on  account 
of  its  high  resistance,  it  is  generally  shunted  by  a  small  inductance, 
or  it  may  be  tapped  across  a  number  of  turns  of  the  inductance 
coil.  The  calibration  is  effected  by  comparing  the  wave  meter 
with  the  standard  wave  meter  in  a  manner  similar  to  that  outlined 
above  for  intercomparing  the  standard  wave-meter  circuits.  The 
condenser  of  the  commercial  instrument  is  set  at  a  given  reading 


Radio  Instruments  and  Measurements 


105 


and  the  wave  length  of  the  source  adjusted  until  the  ammeter  of 
the  wave  meter  indicates  maximum  current.  The  wave  meter  is 
then  detuned,  and  without  changing  the  source  the  standard 
wave  meter  is  adjusted  until  resonance  is  obtained.  The  wave 
length  as  indicated  by  the  standard  corresponds  to  the  chosen 
setting  of  the  commercial  wave  meter,  and,  repeating  the  obser- 
vations for  other  settings,  a  curve  may  be  obtained  giving  the 
wave  length  as  a  function  of  the  setting.  Or  if  it  is  desired  to 
engrave  the  scale  of  the  commercial  instrument  so  as  to  read 
wave  lengths  directly,  the  standard  circuit  may  be  set  at  a  chosen 
integral  wave  length,  the  source  adjusted  to  this  wave  length,  and 
the  corresponding  setting  of  the  commercial  instrument  found  and 


PELATIVE.   AUDIBILITY. 

.85 

55 


FIG.  72. — Various  wave-meter  circuits,  using  detector  and  phones 

marked.  In  these  comparisons  a  source  of  either  damped  or 
undamped  oscillations  may  be  used.  The  latter,  however,  per- 
mits a  higher  precision  in  the  measurement  on  account  of  the 
sharper  tuning.  The  ammeter  of  the  commercial  wave  meter  can 
only  be  used  when  it  is  possible  to  draw  a  considerable  amount  of 
power  from  the  source.  It  is  practically  indispensable  when  the 
wave  meter  is  to  be  used  for  measurements  of  resistance  or  loga- 
rithmic decrement. 

Use  of  Crystal  Detector. — When  the  source  supplies  only  a  small 
amount  of  power,  it  is  necessary  to  use  a  sensitive  indicator,  such 
as  a  crystal  detector  and  phones.  When  such  a  detecting  circuit 
is  connected  or  coupled  to  the  wave-meter  circuit,  the  wave 


io6 


Circular  of  the  Bureau  of  Standards 


length  calibration  and  the  resistance  of  the  wave  meter  will  be 
changed  somewhat,  depending  upon  the  type  of  detecting  circuit. 
The  changes  will  also  depend  to  some  extent  upon  the  adjust- 


FlG.  73. — Increase  of  wave  length  for  different  condenser  settings  due  to 
the  addition  of  detector  circuit 

ment  of  the  crystal  contact,  so  that  it  is  important  in  the  design 
of  a  wave  meter  to  choose  a  detecting  circuit  which  will  least 


0.150 


FlG.   74. — Increase  in  decrement  of  the  wave-meter  circuit  due  to  the 
detector  circuit 

affect  the  wave-meter  constants.  The  wave  meter  should  then 
be  calibrated  with  the  detecting  circuit  connected  as  for  use. 
Circuit  o  in  Fig.  72  represents  the  wave  meter  without  detector, 


Radio  Instruments  and  Measurements 


107 


while   the   circuits   numbered    1-6   show   the   detecting   circuits 
frequently  used. 

Typical  examples  of  the  effects  of  these  circuits  are  shown  in 
Figs.  73  and  74.  Fig.  73  illustrates  the  increased  wave  length 
for  different  condenser  settings  caused  by  the  addition  of  the 
detector  circuit.  The  increase  is  extremely  small  in  the  case  of 
circuits  4  and  6 — i.  e.,  when  the  detecting  circuit  is  connected  to 
the  wave-meter  circuit  at  one  point  or  when  it  is  loosely  coupled 


FlG.  75. — Arrangement  of  circuits  for  compar- 
ing -wave  meters  by  impact  excitation 

to  the  wave-meter  circuit.  In  Fig.  74  the  effect  of  the  detector 
circuit  in  increasing  the  decrement  (or  resistance),  and  hence  in 
impairing  the  sharpness  of  tuning  of  the  wave  meter  is  shown, 
and  here  again  the  circuits  4  and  6  appear  to  produce  the  least 
effect.  As  shown  by  the  relative  audibility  values  in  Fig.  72, 
these  circuits  are  not  as  sensitive  as  those  which  withdraw  more 
power  from  the  main  circuit. 


FlG.  76. — Arrangement  of  circuits  for  deter- 
mining resonance  by  coupling  to  an  aperi- 
odic detector  circuit 

Use  of  Buzzer. — In  order  that  an  audible  note  may  be  heard  in 
the  phone,  the  calibration  must  be  carried  out  using  either  damped 
oscillations  with  a  wave-train  frequency  that  is  audible  or  un- 
damped oscillations  that  are  interrupted  or  "chopped"  at  an 
audible  frequency.  A  simple  and  accurate  method  of  comparison 
is  that  shown  in  Fig.  75.  Here  circuit  /  is  the  buzzer  circuit 
described  on  page  227,  which  excites  the  standard  wave-meter  cir- 


io8 


Circular  of  the  Bureau  of  Standards 


cuit  //  by  impact  excitation.  The  commercial  wave  meter  ///  is 
loosely  coupled  to  the  standard,  and  resonance  is  indicated  by  the 
setting  for  maximum  response  in  the  phones.  Sharpness  in 
setting  is  facilitated  by  reducing  the  coupling  between  //  and  /// 
until  the  phone  responds  only  when  ///  is  very  nearly  in  resonance 
with  //.  The  standard  circuit  has  no  detector  or  buzzer  attached ; 
hence,  its  calibration  is  unaffected  if  it  is  sufficiently  loosely 
coupled  to  /  and  ///.  The  buzzer  circuit  is  generally  not  strictly 
aperiodic  and  will  show  a  very  broad  tuning  at  its  natural  fre- 
quency. It  is  necessary  to  use  it  at  frequencies  differing  con- 
siderably from  its  natural  frequency  or  errors  will  be  introduced. 


FIG.  77. — Arrangement  of  circuits  for  com- 
paring a  wave  meter  with  a  standard  buzzer 
circuit,  the  resonance  point  being  indicated 
by  the  aperiodic  detector  circuit 

Another  method  makes  use  of  a  tuned  buzzer  source,  and  each 
circuit  is  separately  tuned  to  the  source  in  the  same  manner  as 
described  above  for  the  comparison  when  ammeters  are  used. 
Resonance  in  the  standard  circuit  is  indicated  by  an  aperiodic 
detector  circuit  loosely  coupled  to  that  circuit  as  indicated  in 
Fig.  76.  When  the  commercial  wave  meter  is  used  as  a  source, 
the  buzzer  is  usually  connected  as  shown  in  Fig.  77,  circuit  /. 
The  leads  to  the  buzzer  will  add  capacity  to  the  circuit  and  the 
lengths  of  the  waves  emitted  will  be  increased,  in  particular  at  low 
condenser  settings.  The  calibration  is  simply  carried  out  by  the 
circuits  of  Fig.  77.  Circuit  //  is  the  standard  wave  meter  and  /// 
an  aperiodic  detector  circuit  loosely  coupled  to  the  standard  cir- 
cuit. The  coil  of  circuit  ///  may  be  so  oriented  as  not  to  be 
directly  affected  by  circuit  /.  Then  either  circuit  //  may  be  set 


Radio  Instruments  and  Measurements  109 

at  integral  wave  lengths  and  the  settings  found  at  which  these 
wave  lengths  are  emitted  by  /,  or  /  is  set  and  the  wave  length  cor- 
responding to  the  setting  is  found  by  tuning  //. 

CONDENSERS 
31.  GENERAL 

A  condenser  is  an  apparatus  so  designed  that  electrostatic 
capacity  is  its  important  property.  It  consists  of  a  pair  of  con- 
ductors with  their  surfaces  relatively  close  together,  separated  by 
an  insulating  medium  called  the  dielectric.  When  the  two  con- 
ducting plates  are  parallel,  close  together,  and  of  large  area,  the 
capacity  of  a  condenser  is  given  by 

C  =0.0885  Xio 

where  C  is  in  microfarads,  S  =  area  of  one  side  of  one  conducting 
plate  in  cm2,  r  =  thickness  of  dielectric  between  the  plates  in 
centimeters,  and  K,  the  dielectric  constant,  =  i  for  air  and  is  be- 
tween i  and  10  for  most  ordinary  substances.  Formulas  for 
capacities  of  various  combinations  of  conductors,  antennas,  etc., 
are  given  on  pages  237  to  242.  These  formulas  assume  that  the 
charge  is  uniformly  distributed  over  the  surfaces  of  the  conductors, 
no  corrections  being  made  for  edges  or  end  effects.  It  is  seldom 
worth  while,  however,  to  apply  a  correction  on  this  account, 
because  the  capacity  to  the  condenser  case  or  other  conductors 
is  ordinarily  not  calculable,  so  that  the  actual  capacity  of  a  con- 
denser can  be  calculated  only  approximately.  The  actual  value 
is  likely  to  be  in  excess  of  that  calculated.  When  very  accurate 
values  are  required  they  must  be  obtained  by  measurement.  The 
usual  methods  for  measuring  capacities  at  radio  frequencies  are 
discussed  on  pages  129  to  131. 

Series  and  Parallel  Connection.  —  When  two  or  more  condensers 
are  connected  in  series,  the  resultant  capacity  is  given  by 


--  —     —     — 
C    CC2     Cj 

The  resultant  capacity  of  a  number  of  condensers  connected  in 
series  is  always  less  than  the  smallest  capacity  in  the  series.  The 
series  connection  is  used  when  it  is  necessary  to  use  a  voltage 
higher  than  a  single  condenser  would  stand  without  breakdown. 


no  Circular  of  the  Bureau  of  Standards 

When  condensers   are   connected  in  parallel,   their  capacities 
are  simply  added,  thus  : 


The  laws  of  series  and  parallel  combination  of  condensers  are  thus 
the  inverse  of  the  laws  for  resistances.  In  combining  condensers, 
care  must  be  exercised  that  there  are  no  appreciable  mutual 
capacities  between  the  parts  combined. 

Stray  Capacities.  —  It  is  very  difficult  to  concentrate  the  total 
capacity  in  a  radio  circuit  at  a  particular  point  in  the  circuit. 
Every  part  of  the  apparatus  has  capacities  to  other  parts,  and 
these  small  stray  capacities  may  have  to  be  taken  into  account  as 
well  as  the  capacity  of  the  condenser  which  is  intentionally 
inserted  in  the  circuit.  The  stray  capacities  are  particularly 
objectionable  because  they  vary  when  parts  of  the  circuit  or  con- 
ductors near  by  are  moved.  Thus,  they  make  it  difficult  to  keep 
the  capacity  of  the  circuit  constant.  The  disturbing  effects  may 
be  minimized  in  practice,  as  follows:  (i)  Keeping  the  condenser 
a  considerable  distance  away  from  conducting  or  dielectric  masses; 
(2)  shielding  the  condenser,  i.  e.,  surrounding  the  whole  condenser 
by  a  metal  covering  connected  to  one  plate;  (3)  using  a  condenser 
of  sufficiently  large  capacity  so  that  the  stray  capacities  are 
negligible  in  comparison.  The  first  of  these  methods  reduces  only 
the  stray  capacities  of  the  condenser  itself  to  other  parts  of  the 
circuit.  This  is  also  true  of  the  second  method,  which  is  none  the 
less  a  desirable  precaution.  One  of  the  chief  causes  of  variation 
in  the  stray  capacities  is  the  presence  of  the  hand  or  body  of  the 
operator  near  some  point  of  the  circuit.  Shielding  the  condenser 
reduces  the  capacity  variation  from  this  cause.  The  third  method 
is,  in  general,  the  best  for  reducing  or  eliminating  these  errors.  On 
account  of  the  stray  capacities  of  its  various  parts,  the  whole  circuit 
is  in  effect  a  part  of  the  condenser,  and  their  effect  is  best  rendered 
negligible  by  making  the  condenser  capacity  relatively  great. 

Imperfection  of  Condensers.  —  In  an  ideal  condenser,  the  con- 
ductors or  plates  would  have  zero  resistance  and  the  dielectric 
infinite  resistivity  in  all  its  parts.  In  case  an  alternating  emf  is 
applied  to  such  a  condenser,  current  will  flow  into  the  condenser 
as  the  voltage  is  increasing  and  flow  out  as  the  voltage  is  decreasing. 
At  the  moment  when  the  emf  is  a  maximum  no  current  will  be 
flowing,  and  when  the  emf  is  zero  the  current  will  be  a  maxi- 
mum. Hence,  in  a  perfect  condenser  the  current  and  voltage  are 
90°  out  of  phase.  In  actual  condensers  the  conditions  as  to 


Radio  Instruments  and  Measurements 


in 


resistance  in  the  plates  and  dielectric  are  not  fulfilled,  and  in  con- 
sequence an  alternating  current  flowing  in  a  condenser  is  not 
exactly  90°  out  of  phase  with  the  impressed  voltage.  The  differ- 
ence between  90°  and  -the  actual  phase  angle  is  called  the  "phase 
difference.  "  In  an  ideal  condenser  there  would  be  no  consumption 
of  power;  the  existence  of  a  phase  difference  means  a  power  loss, 
which  appears  as  a  production  of  heat  in  the  condenser.  The 
amount  of  the  power  loss  is  given,  as  for  any  part  of  a  circuit, 
by  P  =  EI  cos  6,  where  0  is  the  phase  angle  between  current  and 
voltage  and  cos  6  is  the  power  factor.  This  is  equivalent  to 


where  ^  is  the  phase  difference  and  sin  $  is  the  power  factor  of  the 
condenser.  In  all  except  extremely  poor  condensers,  \l/  is  small, 
sin  \{/  =  \}/,  and  thus  the  phase  difference  and  power  factor  are 
synonymous.  The  power  loss  is  given  by 

P  =  coC£2sin^  (68) 

This  shows  that,  for  constant  voltage,  the  power  loss  is  propor- 
tional to  the  frequency,  to  the  capacity,  and  to  the  power  factor. 
Information  on  the  power  factors  of  condensers  is  given  in 
section  (34)  below. 

Change  of  Capacity  with  Frequency.  —  Another  effect  of  the  im- 
perfection of  dielectrics  is  a  change  of  capacity  with  frequency. 
The  quantity  of  electricity  which  flows  into  a  condenser  during 
any  finite  charging  period  is  greater  than  would  flow  in  during  an 
infinitely  short  charging  period.  In  consequence  the  measured 
or  apparent  capacity  with  alternating  current  of  any  finite  fre- 
quency is  greater  than  the  capacity  on  infinite  frequency.  The 
latter  is  called  the  geometric  capacity  (being  the  capacity  that 
would  be  calculated  from  the  geometric  dimensions  of  the  con- 
denser on  the  assumption  of  perfect  dielectric)  .  The  capacity  of  a 
condenser  decreases  as  the  frequency  is  increased,  approaching 
the  geometric  capacity  at  extremely  high  frequencies.  For  this 
reason,  when  dielectric  constants  are  measured  at  high  frequencies 
of  charge  and  discharge,  smaller  values  are  obtained  than  with 
low  frequencies. 

When  the  phase  difference  of  the  condenser  is  due  to  ordinary 
leakage  or  conduction  through  the  dielectric  or  along  its  surface, 
the  apparent  capacity  at  any  frequency  is  readily  shown  to  be 


35601°—  18 


H2  Circular  of  the  Bureau  of  Standards 

where  C0  is  the  geometric  capacity  in  microfarads,  $  the  phase 
difference,  and  R  the  leakage  resistance  in  ohms.  It  is  evident 
that  the  apparent  capacity  decreases  very  rapidly  as  frequency 
increases.  For  example,  suppose  a  condenser  whose  geometric 
capacity  is  o.ooi  microfarad  to  have  a  leakage  resistance  of  10 
megohms,  the  dielectric  being  otherwise  perfect.  Its  capacity  at 
60  cycles  will  be  0.001070  microfarad,  at  300  cycles  will  be  0.001003 
microfarad,  and  at  all  radio  frequencies  will  be  equal  to  the 
geometric  capacity. 

When  the  phase  difference  of  a  condenser  is  due  to  dielectric 
absorption  (a  phenomenon  discussed  below,  p.  124),  the  capacity 
decreases  as  the  frequency  increases,  as  before,  approaching  the 
geometric  capacity  at  infinite  frequency,  but  the  amount  of  the 
change  can  not  be  predicted  from  a  knowledge  of  the  phase  dif- 
ference. The  change  with  frequency  is  large  in  condensers  that 
have  large  phase  difference.  In  certain  cases  the  change  of  capac- 
ity with  frequency  has  been  found  to  be  roughly  proportional  to 
the  reciprocal  of  the  square  root  of  frequency. 

A  series  resistance  in  the  plates  or  leads  of  a  condenser  causes  a 
phase  difference  but  does  not  give  rise  to  a  change  of  capacity  with 
frequency. 

When  the  leads  inside  the  case  of  a  condenser  are  long  enough 
to  have  appreciable  inductance  the  capacity  measured  at  the 
terminals  appears  to  be  greater  than  it  actually  is.  The  magnitude 
of  the  effect  is  given  by 


where  Ca  is  the  apparent  or  measured  capacity,  and  in  the  paren- 
thesis C  is  in  microfarads  and  L  in  microhenries.  Thus,  the 
inductance  of  the  interior  leads  makes  the  apparent  capacity  of  a 
condenser  increase  as  frequency  increases,  while  the  imperfection 
of  the  dielectric  makes  the  capacity  decrease  with  increase  of 
frequency. 

32.  AIR  CONDENSERS 

Electrical  condensers  are  classified  according  to  their  dielectrics. 
The  plates  are  relatively  unimportant,  their  only  requirement 
being  low  resistance.  This  requirement  is  met  in  the  materials 
used  for  condenser  plates,  viz,  aluminum,  copper,  brass.  When  the 
plates  are  thin  the  material  must  not  have  too  high  a  resistivity. 
Various  dielectrics  are  used;  the  one  most  frequently  used  in 
radio  measurements  is  air. 


Radio  Instruments  and  Measurements  113 

A  condenser  which  is  to  be  used  as  a  standard  of  capacity  in 
measurements  at  radio  frequencies  is  itself  standardized  at  low 
frequencies,  and  its  construction  must  be  such  that  either  the  capac- 
ity does  not  change  with  frequency  or  the  change  can  be  calcu- 
lated. The  capacity  of  a  condenser  with  a  solid  dielectric  changes 
with  the  frequency  in  an  indeterminate  manner,  and  hence  it  is 
practically  impossible  to  calculate  the  capacity  at  high  frequencies 
from  that  measured  at  low  frequency.  Air  is  very  nearly  a  perfect 
dielectric,  hence  a  condenser  with  only  air  as  a  dielectric  should 
show  no  change  in  capacity  with  the  frequency,  and  thus  the 
capacity  at  radio  frequencies  should  be  the  same  as  that  for  low 
frequency.  It  is  on  this  account  that  air  condensers  are  quite 
generally  used  as  standards  of  capacity  in  radio  measurements. 

Phase  Difference  of  Air  Condensers. — Air  condensers  are  valuable 
in  radio  measurements  for  another  reason.  Their  perfection, 
from  a  dielectric  standpoint,  involves  freedom  from  power  loss. 
The  phase  difference  of  an  ideal  air  condenser  is  zero;  there  is  no 
component  of  current  in  phase  with  the  electromotive  force,  and 
thus  the  condenser  acts  as  a  pure  capacity  and  introduces  no 
resistance  into  the  circuit.  It  is  consequently  advantageous  to 
use  them  in  circuits  in  which  it  is  desirable  to  keep  the  resistance 
very  low.  In  resistance  measurements  at  high  frequencies  it  is 
often  necessary  to  assume  that  the  resistance  of  the  condenser  in 
the  circuit  is  negligible.  This  requires  the  use  of  a  properly  con- 
structed air  condenser. 

Only  the  most  careful  design,  however,  can  produce  an  air 
condenser  which  is  close  to  perfection.  In  order  to  support  the 
two  conductors  or  sets  of  plates  and  insulate  them  from  each 
other  it  is  necessary  to  introduce  some  solid  dielectric.  There  is 
necessarily  some  capacity  through  this  dielectric,  and  since  all 
solid  insulators  are  imperfect  dielectrics  this  introduces  a  phase 
difference.  The  magnitude  of  the  phase  difference  is  determined 
by  the  quality  of  the  solid  dielectric  and  the  relative  capacities 
through  this  dielectric  and  through  the  air.  The  effect  is  magni- 
fied by  the  concentration  of  the  lines  of  electric  field  intensity  in 
solid  dielectric  due  to  its  high  dielectric  constant.  Some  air  con- 
densers tested  by  this  Bureau,  in  which  the  pieces  of  dielectric 
used  as  insulators  were  large  and  poorly  located,  had  phase  differ- 
ences or  power  factors  many  times  greater  than  those  of  commercial 
paper  condensers.  Although  purporting  to  be  air  condensers  they 
were  actually  poorer  than  ordinary  solid-dielectric  condensers 


ii4  Circular  of  the  Bureau  of  Standards 

because  the  insulating  pieces  used  to  separate  the  plates  were  very 
poor  dielectrics.  In  a  variable  air  condenser  the  phase  difference 
varies  with  the  setting  and  is  approximately  inversely  proportional 
to  the  capacity  at  any  setting.  The  equivalent  resistance  (defined 
on  p.  125)  is  inversely  proportional  to  the  square  of  capacity  at  any 
setting. 

The  questions  of  materials  and  construction  of  air  condensers 
are  further  dealt  with  below  in  connection  with  design.  It  is 
seldom  safe  to  assume  that  an  air  condenser  has  zero  phase 
difference. 

Antennas  are  subject  to  this  same  imperfection.  An  antenna 
is  essentially  an  air  condenser  and  is  similarly  subject  to  power 
loss  from  poor  dielectrics  u  in  its  field. 

Simple  Variable  Condenser.  —  In  the  most  generally  used  types 
of  air  condensers  the  capacity  is  continuously  variable.  Variable 
condensers  are  extensively  used,  because  most  radio  measure- 
ments involve  a  variation  of  either  inductance  or  capacity,  and 
it  is  relatively  difficult  to  secure  sufficient  variation  of  an  induct- 
ance without  variation  of  the  resistance  and  capacity  in  the  cir- 
cuit. The  most  familiar  type  of  variable  air  condenser  has  two  sets 
of  semicircular  plates  (see  Fig.  78  ,  facing  p.  1  1  8)  ,  one  set  of  which  can 
be  revolved,  bringing  the  plates  in  or  out  from  between  the  plates 
of  the  fixed  set.  The  position  of  the  movable  plates  is  indicated 
by  a  pointer  moving  over  a  scale  which  is  marked  off  in  arbitrary 
divisions.  These  may  be  degrees,  o°  corresponding  to  the  posi- 
tion of  the  plates  when  they  are  completely  outside  of  the  fixed 
set  and  the  capacity  is  a  minimum,  and  180°  when  the  capacity 
is  a  maximum.  It  is  preferable  to  divide  the  range  into  100 
divisions  rather  than  into  degrees.  The  capacity  of  a  condenser 
is  proportional  to  the  area  of  the  plates.  In  a  variable  condenser 
of  the  semicircular  type  the  effective  area  of  the  plates  is  changed 
by  rotating  the  movable  plates  and,  neglecting  the  edge  effects, 
it  is  changed  in  proportion  to  the  angle  of  rotation.  As  a  result 
the  capacity  is  approximately  proportional  to  the  setting  through- 
out a  wide  range,  provided  that  the  condenser  is  well  constructed 
and  the  distance  between  the  two  sets  of  plates  is  not  affected 
by  rotation  of  the  movable  set.  Fig.  79  shows  a  typical  capacity 
curve  for  such  a  condenser.  Throughout  the  range  in  which  the 
capacity  curve  is  a  straight  line  the  capacity  is  given  by  the 
formula 


11  See  reference  No.  197,  Appendix  2. 


Radio  Instruments  and  Measurements 

where  a  and  b  are  constants  and  6  is  the  setting  in  scale  divisions. 
The  coefficient  a  represents  the  change  in  capacity  for  one  division 
and  may  be  computed  by  taking  the  difference  of  the  capacities 
at,  say,  20  and  80  and  dividing  by  60.  The  constant  b  represents 
the  capacity  which  the  condenser  would  have  at  o  if  its  linear 
character  were  maintained  down  to  this  setting.  It  may  be 
positive,  negative,  or  zero,  depending  upon  the  setting  of  the 
pointer  relative  to  the  movable  plates.  Its  value  is  determined 
by  subtracting  the  value  of  aff  at,  say,  30  from  the  actual  value 
of  the  capacity  at  that  setting.  When  a  condenser  has  a  capacity 
curve  that  is  closely  linear,  it  may  be  found  easier  to  compute 


2800 
2600 
240O 
2200 
2000 
(800 
1600 
MOO 
I2OO 

toco 

/ 

/ 

A 

/ 

/ 

/ 

/ 

/ 

< 

/ 

< 

/ 

5 

a 

/ 

2 

/ 

/ 

COO 

/ 

/ 

/ 

/ 

o 

y 

Sc< 

-r  r 

VISIO 

*s 

10        20      JO       4O       iO       60       70       80       90      ICO 

FIG.  79. — Typical  capacity  curve  for  con- 
denser with  semicircular  plates 

the  capacity  for  given  settings  by  means  of  the  formula  rather 
than  to  read  off  the  values  from  a  curve. 

Uniform  Wave-Length  Type. — Some  condensers  have  been  spe- 
cially designed  to  give  a  capacity  curve  different  from  the  cus- 
tomary linear  curve.  In  a  wave  meter  in  which  a  semicircular 
plate  condenser  is  used,  for  any  one  coil  the  wave  length  varies 
as  the  square  root  of  the  capacity  and  hence  as  the  square  root 
of  the  setting  of  the  condenser.  If  it  is  attempted  to  make  the 
wave  meter  direct  reading — i.  e.,  to  substitute  a  wave-length 
scale  in  place  of  the  scale  of  setting  in  degrees — this  wave-length 
scale  will  be  nonuniform  and  either  crowded  together  too  closely 
at  the  low  settings  or  too  open  at  the  high  settings.  In  order 
to  obtain  a  uniform  scale  of  wave  lengths,  it  is  necessary  to  have 


n6 


Circular  of  the  Bureau  of  Standards 


the  capacity  vary  as  the  square  of  the  displacement  or  rotation. 
Tissot 12  proposed  a  condenser  which  had  two  sets  of  square 
plates  (Fig.  80)  which  moved  relative  to  each  other  along  their 
diagonals.  The  same  result  can  be  obtained  in  a  rotary  con- 
denser if  one  set  of  plates  is  given  the  proper  shape.13  It  is 
required  that  the  capacity  and,  hence,  the  effect- 
ive area  between  the  plates  shall  vary  as  the 
square  of  the  angle  of  rotation.  Thus, 


FIG.  80. — Form  of 
condenser  plates 
for  which  the  ca- 
pacity varies  as  the 
square  of  the  dis- 
placement 


But  in  polar  coordinates  the  area  is  equal  to 

-!/• 


. 


2  ™ 


Differentiating  these  two  values  of  A, 

dA     i 


r  = 


In  the  condenser  as  actually  made  the  fixed  plates  may  be 
semicircular  and  the  moving  plates  given  the  required  shape  to 


FIG.  81. — Form  of  rotary  condenser  plates 
for  which  the  capacity  varies  as  the  square 
of  angular  displacement 

make  the  effective  area  vary  as  the  square  of  the  angle  of  rotation. 
This  effective  area  is  the  projection  of  the  moving  plates  on  the 
fixed.  To  provide  clearance  for  the  shaft  of  the  moving-plate 
system  a  circular  area  of  radius  r2  must  be  cut  from  the  fixed 


12  See  reference  No.  81,  Appendix  2. 


13  See  reference  No.  82,  Appendix  2. 


Radio  Instruments  and  Measurements  117 

plates,  and,  taking  this  into  account,  the  equation  of  the  boundary 
curve  of  the  moving  plates  becomes 


In  Fig.  8  1  the  form  of  the  plates  is  shown  and  the  effective  area 
denoted  by  shading. 

Decremeter  Type.  —  Another  special  shape  of  plates  is  utilized  in 
the  direct-reading  decremeter  14  developed  at  this  Bureau.  As 
shown  in  section  55,  logarithmic  decrement  may  be  measured  by 
the  per  cent  change  in  capacity  required  to  reduce  by  a  certain 
amount  the  indication  of  an  instrument  in  the  circuit  at  resonance. 
In  order  that  equal  angular  rotations  may  correspond  to  the 
same  decrement  at  any  setting  of  the  condenser,  it  is  necessary 
that  the  per  cent  change  in  capacity  for  a  given  rotation  shall  be 

the  same  at  all  parts  of  the  scale.     Thus  we  have  the  requirement 
j/"* 

a  =  constant  =  per  cent  change   of   capacity  per   scale 


division.     By  integration, 

log  C  =  ad  +  b  where  b  =  a  constant,  or 

C  = 


FIG.  82  .  —  Form  of  rotary  condenser  plates  in 
which  the  per  cent  change  of  capacity  is 
the  same  throughout  the  entire  range  of 
the  condenser 


C0  =  e6  =  capacity  when  6  =  O. 

Since  the  area  must  vary  as  the  capacity 


A=- 

2 

—  =  ~  =  r     <* 

dd~2  ~L°ae 
r  =  J'2Crfrt*  (69) 


M  See  reference  No.  196,  Appendix  2,  and  description  of  decremeter  on  p.  199- 


1  1  8  Circular  of  the  Bureau  of  Standards 

This  latter  is  then  the  polar  equation  of  the  bounding  curve 
required  to  give  a  uniform  decrement  scale. 

The  shape  of  the  condenser,  as  actually  made,  is  shown  in  Fig. 
82  and  Fig.  218,  facing  page  320.  A  small  semicircular  area  is 
omitted  from  the  fixed  plates,  to  provide  clearance  for  the  metal 
washers  which  must  hold  the  moving  plates  together.  Taking 
account  of  this  omitted  area,  of  radius  r2,  the  equation  of  the 
boundary  curve  becomes 

r  = 


The  effective  area  is  denoted  by  shading  in  Fig.  82. 

A  wave-length  scale  placed  on  such  a  condenser  is  somewhat 
crowded  at  high  settings  —  just  the  opposite  of  the  effect  with  a 
semicircular  plate  condenser.  It  is  much  more  nearly  uniform 
than  in  the  case  of  a  semicircular-plate  condenser,  as  might  be 
expected  from  the  similarity  of  shape  of  Figs.  81  and  82. 

Important  Points  in  Design.  —  In  a  standard  condenser  it  is 
required  that  the  capacity  remain  constant  and  be  definite.  The 
former  condition  requires  rigidity  of  construction,  which  is  diffi- 
cult to  secure  in  a  variable  condenser.  The  pointer  and  movable 
plates  must  be  securely  fastened  to  the  shaft  so  that  no  relative 
motion  is  possible.  A  simple  set  screw  is  not  sufficient  to  hold 
the  pointer  in  place.  It  is  preferable  to  have  no  stops  against 
which  the  pointer  may  hit.  Particular  care  must  be  exercised  in 
insulating  the  fixed  and  moving  plates  from  each  other.  The  sus- 
pension of  heavy  sets  of  plates  from  a  material  such  as  hard 
rubber,  which  may  warp,  is  objectionable.  In  some  cases  the 
high  temperature  coefficient  of  expansion  of  insulators  may  pro- 
duce relative  motions  of  the  two  sets  of  plates,  resulting  in  a  high 
temperature  coefficient  of  capacity.  In  order  to  make  the  capac- 
ity definite  and  also  minimize  power  loss,  it  is  desirable  to  sur- 
round the  condenser  with  a  metal  case  which  is  connected  to 
one  set  of  plates  and  grounded  when  the  condenser  is  calibrated 
and  used.  The  inductance  of  the  leads  inside  of  the  condenser 
should  be  a  minimum,  for  the  apparent  capacity  of  a  condenser 
at  high  frequencies  will  increase  with  the  frequency  due  to  induc- 
tance in  the  leads  in  a  similar  manner  to  the  variation  of  apparent 
inductance  of  a  coil  with  distributed  capacity.  The  connections 
from  the  binding  posts  to  the  plates  should  therefore  be  short  and 
thick;  this  minimizes  both  inductance  and  resistance. 


Bureau  of  Standards  Circular  No.  74 


FlG.  78. — Commercial  types  of  variable  air  condensers 


FIG.  84. — Quartz-pillar  standard  condenser  of        FIG.   85. — Type  of  -variable    con- 
fixed value  denser  suitable  for  high  voltages, 

with  double  set  of  semicircular 
moving  plates 


Bureau  of  Standards  Circular  No.  74 


FIG.  86. — Leyden  jar  type  of  high  voltage  condenser 


FIG.  87. — Mica  condensers  suitable  for  high  voltages 


Radio  Instruments  and  Measurements 


119 


The  resistance  of  leads  and  plates  and  contact  resistances 
between  the  individual  plates  and  separating  washers  should  be 
kept  as  low  as  possible,  in  order  to  minimize  the  phase  angle. 
For  the  same  reason  the  dielectric  used  to  support  one  set  of 
plates  and  insulate  the  two  sets  from  each  other  should  be  as 
nearly  perfect  as  possible.  The  insulation  resistance  must  be 
high,  because  otherwise  it  would  introduce  an  error  in  the  meas- 
ured capacity  at  low  frequencies  (see  p.  in).  Power  loss  in  the 


ELEVATION 

SHOWING  CASE  BROKEN  AWAV 

FIG.  83. — Quartz-pillar  variable  condenser  used  as  primary  standard  of  capacity  at 

high  frequencies 

dielectric  should  be  kept  small  by  using  suitable  insulating  ma- 
terial, locating  it  where  the  electric  field  intensity  is  not  great,  and 
using  pieces  of  such  size  and  shape  that  the  capacity  through  it 
is  small.  Built-up  mica,  bakelite,  formica,  and  similar  materials 
have  been  found  to  have  bad  dielectric  losses.  Hard  rubber  and 
porcelain  are  better.  For  further  information  on  dielectrics,  see 
section  34  below,  on  Power  Factor.  In  special  cases  quartz  may 
be  used,  as  in  the  special  condensers  next  described. 


1 20  Circular  of  the  Bureau  of  Standards 

Bureau  of  Standards  Type. — In  the  customary  design  of  variable 
air  condensers,  the  movable  plates  are  insulated  from  the  fixed 
by  rings  of  insulating  material  around  the  bearings  at  top  and 
bottom.  This  introduces  some  difficulties  in  the  design  of  the 
bearing;  the  capacity  through  the  dielectric  is  likely  to  be  large 
and  the  choice  of  insulator  is  limited. 

In  the  condensers  used  as  standards  at  this  Bureau,  instead  of 
the  moving  plates  being  insulated  from  the  case  of  the  condenser, 
they  are  in  electrical  connection  with  the  case  and  the  bearings 
are  metal.  The  fixed  plates  are  insulated  from  the  moving  plates 
and  the  case  by  pieces  of  quartz  rod  inserted  in  the  uprights  which 
support  the  fixed  plates.  The  capacity  through  the  insulator  is 
small.  The  use  of  quartz  is  desirable  on  account  of  very  high 
insulation  and  low  temperature  coefficient  of  expansion.  The 
condensers  of  this  type  which  have  been  tested  have  shown  per- 
manence, no  measurable  power  loss,  and  a  very  low  temperature 
coefficient  of  capacity.  On  account  of  the  brittleness  of  the 
quartz  and  the  heavy  weight  supported  by  it,  these  condensers 
are  not  very  portable  and  must  be  dismounted  in  shipping. 
Views  of  the  inside  and  the  case  of  these  condensers  are  shown 
in  Fig.  214,  facing  page  318.  Two  sizes,  of  0.007  and  0.0035 
microfarad  capacity,  are  shown.  Data  on  the  construction  are 
given  in  Fig.  83. 

Fixed  Condensers. — Air  condensers  in  which  the  two  sets  of 
plates  are  fixed  relative  to  each  other  are  particularly  valuable 
when  well  constructed  on  account  of  their  permanence.  They 
serve  to  standardize  variable  air  condensers,  which  are  more  likely 
to  change,  and  are  of  value  in  standardizing  solid-dielectric  con- 
densers. In  Fig.  84,  facing  page  1 18,  is  shown  a  type  designed  at 
this  Bureau,  of  o.oi  microfarad  capacity,  which  has  two  sets  of 
square  plates  and  which  is  insulated  by  quartz  supports  in  a  man- 
ner similar  to  the  variable  condensers  just  described.  Great  care 
is  necessary  in  the  construction  and  maintenance  of  these  con- 
densers. The  brass  plates  must  be  thoroughly  annealed;  the  air 
must  be  kept  quite  dry. 

33.  POWER  CONDENSERS 

The  condensers  used  in  radio  transmitting  circuits  carry  large 
amounts  of  power  and  are  called  power  condensers.  They  are 
operated  at  high  voltages,  and  this  requires  special  construction. 
The  voltage  required  in  transmitter  use  may  be  found  as  follows : 


Radio  Instruments  and  Measurements  121 

The  power  input  is  measured  by  the  energy  stored  in  the 
capacity  at  each  charge  multiplied  by  the  number  of  charges  or 
discharges  per  second.  Thus  P  =  y^CE^N  where  N  is  the  number 
of  discharges  per  second,  or  the  spark  frequency. 

2p 
Hence  EQ2  =  - 


In  a  typical  case  N  =  1000,  C  =  o.oi6  X  io~6  farad  and  P  =  2ooo 
watts;  then 


/  2  (2i 

'°~  V  1000(0. 


=  1  6000  volts. 


016)10' 

Air  and  Oil  Condensers. — The  foregoing  indicates  the  order  of 
the  voltage  requirement  for  power  condensers.  In  the  case  of  air 
condensers  of  ordinary  construction  this  high  voltage  would  neces- 
sitate very  large  spacing  between  the  plates,  and  this,  in  turn, 
would  necessitate  a  very  big  volume  for  a  moderate  capacity. 
If,  however,  the  air  condensers  have  a  strong  air-tight  case,  so 
that  the  air  inside  the  case  can  be  compressed  up  to  15  or  20 
atmospheres,  the  breakdown  voltage  becomes  very  high,  even  for 
a  small  distance  between  the  plates.  Thus  voltages  as  high  as 
35  ooo  may  be  used  with  3 -mm  spacing  between  the  plates.  At 
the  same  time  the  brush  discharge  losses  are  reduced  so  as  to  be 
negligible.  Such  condensers  have  the  advantage  of  low  power 
loss,  but  the  disadvantage  of  being  quite  bulky.  A  size  com- 
monly used  has  a  capacity  of  about  0.005  microfarad.  Another 
way  of  utilizing  the  air  condenser  type  for  high  voltages  is  to  fill 
the  condenser  with  oil.  Using  ordinary  petroleum  oils,  this  about 
doubles  the  capacity.  Condensers  with  oil  as  a  dielectric  are  very 
well  suited  for  power  condensers.  The  breakdown  voltage  is  very 
high,  dielectric  and  brush  losses  low,  and  on  account  of  the  high 
dielectric  constant  of  some  oils,  it  is  easy  to  get  a  large  capacity 
in  a  moderate  volume.  A  variable  condenser  suitable  for  oil 
immersion  is  shown  in  Fig.  85,  facing  page  118.  Only  condensers 
with  liquid  or  gas  dielectrics  can  have  continuously  variable 
capacity.  This  is  a  great  advantage,  and  such  a  condenser  as  this 
is  a  most  valuable  supplement  to  the  fixed  condensers  of  solid 
dielectric  used  in  a  high-voltage  circuit,  since  it  makes  possible 
easy  variation  of  the  wave  length. 

Glass  Condensers. — Condensers  with  glass  for  a  dielectric,  and 
especially  the  cylindrical  Leyden  jars  with  copper  coatings,  are 
very  extensively  used  for  power  condensers.  Special  types  have 


122  Circular  of  the  Bureau  of  Standards 

been  devised  to  increase  breakdown  voltage  and  reduce  brush 
discharge.  This  is  accomplished  with  the  ordinary  types  by 
immersion  in  oil,  which  also  improves  the  insulation.  Glass  con- 
densers are  relatively  cheap.  They  must,  of  course,  be  carefully 
handled.  A  stock  size  of  Ley  den  jar  has  a  capacity  of  0.002 
microfarad.  The  power  factor  of  glass  condensers  is  rather  large 
(see  next  section).  When  used  in  the  primary  circuit  of  a 
quenched  gap  transmitter,  however,  moderate  power  loss  in  the 
condensers  is  probably  not  important  in  reducing  the  efficiency  of 
the  set  since  the  power  losses  in  the  rest  of  the  circuit  are  high 
and  the  circuit  is  operative,  if  proper  quenching  is  secured,  for  but 
a  small  fraction  of  the  time.  (See  Fig.  86,  facing  p.  119.) 

Mica  Condensers. — These  are  coming  into  use  to  a  considerable 
extent  in  radio  work,  both  as  power  condensers  and  as  standards. 
They  have  the  advantages  of  low  power  loss,  small  volume,  and 
are  not  fragile.  Stock  sizes  include  0.002  and  0.004  microfarad. 
(See  Fig.  87,  facing  p.  119.)  In  order  to  withstand  high  voltages, 
the  mica  sheets  are  comparatively  thick  and  several  sections  are 
joined  in  series.  When  they  are  properly  made  and  have  very 
small  phase  difference,  they  may  be  used  as  standard  condensers ; 
and  very  conveniently  supplement  the  air  condensers  ordinarily 
used  as  standards,  being  obtainable  in  larger  capacities.  They  are 
valuable  as  standards  both  on  account  of  their  permanence  and  the 
large  capacity  obtainable  in  a  small  volume,  but  must  be  standard- 
ized both  in  respect  to  capacity  and  power  factor  for  the  range  of 
frequencies  at  which  they  are  to  be  used.  While  the  capacity  and 
power  factor  may  be  considerably  different  at  high  frequencies  from 
the  values  at  low  frequency,  it  is  fortunate  that  throughout  the  range 
of  radio  frequencies  both  of  these  quantities  are  practically  con- 
stant. An  exception  must  be  made  in  the  case  of  the  power 
factor  if  any  considerable  portion  of  the  power  loss  is  due  to 
ohmic  resistance  in  the  leads  or  plates;  for  under  these  circum- 
stances, the  power  factor  will  increase  with  increasing  frequency, 
as  pointed  out  below.  In  properly  constructed  condensers,  how- 
ever, power  loss  from  this  source  is  negligible. 

34.  POWER  FACTOR 

The  power  loss  in  a  condenser  may  be  due  either  to  imperfection 
of  the  dielectric  or  to  resistance  in  the  metal  plates  or  leads.  The 
dielectric  may  cause  a  power  loss  either  by  current  leakage,  by 
brush  discharge,  or  more  commonly  by  the  phenomenon  described 
later  under  the  head  of  "  Dielectric  absorption." 


Radio  Instruments  and  Measurements 


123 


Leakage. — The  leakage  of  electricity  by  ordinary  conduction 
through  the  dielectric  or  along  its  surface  contributes  to  the  phase 
difference  at  low  frequencies  but  is  generally  negligible  at  high 
frequencies.  The  effect  of  leakage  on  the  power  factor  may  be 
seen  as  follows:  A  condenser  having  leakage  may  be  represented  by 
a  pure  capacity  with  a  resistance  in  parallel.  The  current  divides 
between  the  two  branches,  the  current  /R  through  the  resistance 
being  in  phase  with  the  applied  E,  and  the  current  Io  through  the 
capacity  leading  E  by  90°.  The  resultant  /  leads  E  by  an  angle 
which  is  less  than  90°  by  the  phase  difference  \f/.  From  Fig.  88, 

T 

tan  $  = 


RwC 

The  effect  of  R  may  be  shown  by  an  example.     A  condenser 
of  o.oi  microfarad  capacity  with  an  insulation  resistance  as  low 

R 


AAAAMM 


FIG.  88. — Equivalent  circuit  and  -vector  diagram 
for  condenser  having  leakage 

as  10  megohms  has,  at  a  frequency  of  60  cycles  per  second,  a 

power  factor  =  - — — — r —  =  0.027  =  2.7   per  cent.      This  is  a 

(io)7  377  (10)-" 

very  appreciable  quantity;  2.7  per  cent  of  the  current  flows  by 
conduction  instead  of  by  dielectric  displacement.  This  effect, 
however,  decreases  as  the  frequency  increases,  for  the  dielectric 
current  increases  in  proportion  to  the  frequency  while  the  leak- 
age current  does  not,  For  instance,  at  io  ooo  cycles,  the  power 
factor  =  0.0001 6.  Thus  at  radio  frequencies  the  power  factor 


124  Circular  of  the  Bureau  of  Standards 

due  to  conduction  through  any  but  an  extremely  poor  condenser 
is  wholly  negligible. 

Series   Resistance. — A  resistance  within  a  condenser  in  series 
with  the  capacity  affects  the  power  factor  very  differently  from 
a  resistance  in  parallel.     The  series  resistance  includes  the  resist- 
ance of  plates,  joints,  or  contacts,  and  the  leads  from  binding 
posts  to  plates.     The  Et  across  the  resistance  is  in  phase  with  the 
current  /,  and  the  emf  Eo  across  the  capacity  is  90°  behind  /  in 
phase.     The  power  f actor  =  sin  $,  and  since  ^  is  usually  small,  it 
may  be  taken  as  =  tan  \l/,  which  from  Fig.  89  is  rcoC. 

If  r  =  i  ohm  and  C=o.oi  microfarad  the  power  factor  at  60 
cycles  =  3. 8    (io)~6.     This  is  utterly  negligible.     However,    at   a 
frequency  of  i  ooo  ooo  cycles,  the  power  factor  =  0.063  =  6. 3  per 
cent,  which  is  so  large  as  to  be  serious.     Thus,  it  is  important  to 

minimize  series  resistance  in  condens- 

— 1[£— /ty\/yUAAAA__     ers   for   radio   work,   while  condenser 
leakage  on  the  other  hand  has  its  chief 
_______^_      importance  at  low  frequencies. 

In  the  foregoing  example,  r  was  taken 
as  i  ohm.      In  actual  condensers  it  is 
sometimes  greater  than  this.     A  plate 
resistance  of  several  ohms  is  common 
in  the  ordinary  paper  condenser.     In 
FIG.  89. — Equivalent  circuit  and  most  other  condensers,  a  high  series  re- 
vector    diagram  for    condenser    sistance  indicates  a  defect. 
having  dielectric  losses  or  plate         ^  .     ,  .         ~, 

resistance  ^n  account  of  skin  effect,  the  series 

resistance  in  a  condenser  increases  to 

some  extent  with  frequency.  In  consequence,  the  power  factor 
increases  in  proportion  to  a  power  of  the  frequency  slightly  greater 
than  unity.  The  magnitude  of  this  effect  in  typical  condensers  is 
not  known. 

Dielectric  Absorption. — When  a  condenser  is  connected  to  a 
source  of  emf  such  as  a  battery,  the  instantaneous  charge  is  fol- 
lowed by  the  flow  of  a  small  and  steadily  decreasing  current  into 
the  condenser.  The  additional  charge  seems  to  be  absorbed  by 
the  dielectric.  Similarly,  the  instantaneous  discharge  of  a  con- 
denser is  followed  by  a  continuously  decreasing  current.  It 
follows  that  the  maximum  charge  in  a  condenser  cyclically  charged 
and  discharged  varies  with  the  frequency  of  charge.  The  phe- 
nomenon is  similar  to  viscosity  in  a  liquid,  and  is  sometimes 
called  "  dielectric  viscosity.  " 


Radio  Instruments  and  Measurements 


125 


Dielectric  absorption  is  always  accompanied  by  a  power  loss, 
which  appears  as  a  production  of  heat  in  the  condenser.  The 
existence  of  a  power  loss  signifies  that  there  is  a  component  of 
emf  in  phase  with  the  current.  The  effect  of  absorption  is  thus 
equivalent  to  that  of  a  resistance  either  in  series  or  in  parallel 
with  the  condenser.  It  is  found  most  convenient  to  represent 
absorption  in  terms  of  a  series  resistance,  which  is  spoken  of  as 
the  "equivalent  resistance"  of  the  condenser.  An  absorbing 
condenser  is,  therefore,  considered  from  the  standpoint  of  Fig. 
89,  and  the  power  factor  =  rcoC.  The  equivalent  resistance  r 


z 
q 

u 
u 

I 

V) 

u 
tt 


CO 

JIV 

AL- 

E:N 

r  f 

?E£ 

\s- 

~f\t 

ICE 

/ 

OF 

GU 

AS 

sc 

ON 

DE 

*s 

CR 

> 

J 

CAI 

»A 

:IT 

Y' 

=  0. 

501 

KI 

if 

/ 

/ 

/ 

/ 

/ 

/ 

/ 

/ 

' 

-»• 

/ 

/ 

/ 

/ 

/ 

f 

j 

/ 

/ 

v 

• 

1000  2000  3000 

WAVE  LENGTH  .METERS 

FIG.  90. — Variation  of  dielectric  loss  in  glass  with  wave  length 

is  constant  for  a  given  frequency  but  is  different  for  different 
frequencies. 

The  variation  of  the  power  factor  and  the  equivalent  resistance 
with  frequency  is  a  complicated  matter,  the  laws  of  which  are 
not  accurately  known.  To  a  first  approximation,  however,  the 
power  factor  of  an  absorbing  condenser  is  constant.  Since  rcoC 
is  approximately  constant,  r  is  inversely  proportional  to  fre- 
quency (and  therefore  directly  proportional  to  wave-length). 
This  is  well  shown  by  the  nearly  straight  line  in  Fig.  90;  the 


126 


Circular  of  the  Bureau  of  Standards 


crosses  show  observed  resistances  for  a  glass  condenser.  For 
most  condensers  used  in  radio  circuits,  the  power  factor  is  con- 
stant over  the  range  of  radio  frequencies  and  has  nearly  the  same 
value  at  low  frequencies.  This  fact,  viz,  that  power  factor  or 
phase  difference  is  approximately  independent  of  frequency,  is 
very  convenient  and  easily  remembered. 

The  same  law  holds  for  antennas  at  frequencies  less  than  those 
for  which  the  radiation  resistance  is  large;  the  equivalent  resistance 
is  inversely  proportional  to  frequency.  Otherwise  expressed,  the 
equivalent  resistance  is  proportional  to  wave  length,  for  wave 
lengths  greater  than  those  at  which  radiation  is  appreciable.  This 
is  shown  by  Fig.  91,  and  is  discussed  on  page  81.  It  is  believed 


-5  -j 


600  1000  1 500  ZCOO  2600  3OOO 

WAVE  LENGTH,  METERS 
FIG.  91. — Variation  of  antenna  resistance  with  wave  length 

to  be  due  to  the  presence  of  imperfect  dielectrics  (such  as  build- 
ings, insulators,  and  trees)  in  the  field  of  the  antenna;  thus  an  an- 
tenna is  like  an  air  condenser  in  respect  to  the  effect  of  poor  dielec- 
trics in  its  field,  as  well  as  in  other  ways. 

Values  of  power  factor  and  equivalent  resistance  are  given  for 
typical  radio  condensers  in  Table  2.  The  power  factor  does  not 
vary  with  the  size  of  the  condenser;  it  is  a  function  of  the  dielectric 
and  not  of  the  particular  condenser.  The  equivalent  resistance, 
on  the  other  hand,  is  inversely  proportional  to  the  capacity  of  the 
condenser,  since  power  factor  =  ruC.  The  data  in  the  table  for 
Ley  den  jars  in  oil  are  really  typical  of  the  glass  dielectric,  while  the 


Radio  Instruments  and  Measurements 


127 


relatively  large  power  factor  given  for  Ley  den  jars  in  air  is  due  to 
brush  discharge.  The  brush  discharge  is  a  function  of  voltage  (see 
Brush  Loss  below),  and  is  only  appreciable  at  voltages  above 
10  ooo.  For  all  voltages  below  this,  the  power  factor  of  a  Ley  den 
jar  in  air  is  about  the  same  as  the  value  given  for  a  Ley  den  jar  in 
oil,  about  0.003. 

TABLE  2.— Power  Factors  of  Radio  Condensers  «  (at  14  500  volts) 


Kind  of  condenser 

Power  factor 

Capacity 

Equivalent 
resistance 
at  1000  m 

Compressed  air'*  

0.001 

0.0058 

0.14 

.003 

.0060 

28 

Molded  (Murdock),  new  

.004 

.0054 

.41 

Glass  plates  in  oil               

.005 

.0042 

.58 

Glass  (Moscicki  type)            

.006 

.0055 

57 

Glass  (Leyden  jar)  in  air              

.016 

.0061 

1  4 

Molded  micanite       

.023 

0041 

2  9 

Paper  

.024 

.0058 

2.2 

o Calculated  from  determinations  by  Austin,  Bull.  B.  S.,  9,  p.  77;  1912. 

&  The  power  factor  observed  for  the  compressed-air  condenser  is  attributable  not  to  the  air  but  to  the 
resistance  in  its  leads  and  plates,  to  eddy  current  loss  in  the  metal  case,  and  to  the  insulating  material 
used  to  separate  the  two  sets  of  plates. 

Except  for  the  compressed-air  condenser  and  the  Leyden  jar  in 
air,  the  power  factors  do  not  vary  with  voltage  nor  to  a  great 
extent  with  frequency.  The  power  factors  of  solid  dielectrics 
are  principally  due  to  dielectric  absorption.  If  an  appreciable 
portion  of  the  power  factor  were  due  to  a  series  resistance,  such  as 
a  high  plate  resistance,  it  would  be  manifested  by  an  increase  of 
power  factor  with  increasing  frequency.  At  frequencies  higher 
than  300000  (corresponding  to  a  wave  length  of  1000  meters), 
this  is  to  be  expected  in  the  case  of  paper  condensers,  which  have 
long  tin-foil  plates;  the  power  factor  will  increase,  and  the  equiva- 
lent resistance  can  not  decrease  to  a  value  lower  than  the  plate 
resistance.  When  the  power  factor  is  due  purely  to  dielectric 
absorption  the  equivalent  resistance  decreases  in  proportion  to  an 
increase  of  frequency  (or  increases  in  proportion  to  an  increase  of 
wave  length) . 

The  dependence  of  power  loss,  power  factor,  and  equivalent 
resistance  upon  the  frequency,  in  the  case  of  absorbing  condensers, 
condensers  with  leakage,  etc.,  is  summarized  in  the  following  table. 
P  is  the  power  loss,  r  the  equivalent  series  resistance,  \f/  the  phase 
difference  (same  as  power  factor) ,  o>  the  usual  2ir  times  frequency, 

35601°— 18 9 


128 


Circular  of  the  Bureau  of  Standards 


and  X  wave  length.     Much  of  the  information  in  the  table  is  con- 
tained in  the  equations 


The  table  shows  the  powers  of  frequency  and  wave  length  to 
which  P,  \f/,  and  r  are  proportional.  An  arbitrary  notation  is 
used  in  the  exponents  of  w  and  X;  the  symbols  >  and  <  for 
"greater  than"  and  "less  than"  are  used  to  indicate  the  mag- 
nitudes of  the  powers. 

TABLE  3.  —  Variation  of  Phase  Difference,  etc.,  with  Frequency 


Kind  of  system 

1 

-> 

I' 

1 

Condenser  with  leakage  

w° 

x° 

W-1 

X1 

w-2 

X2 

Absorbing  condenser  at  very 
low  frequencies  

w<! 

X>-J 

o><° 

X>° 

oK-1 

X>* 

Absorbing  condenser  at  radio 
frequencies  

w1 

X-1 

w° 

X° 

w-1 

X1 

Constant  series  resistance  
Skin  effect  in  conductors  

CO2 

w>2 

X-2 
X<-2 

w1 
w>! 

X-1 

x<-1 

o>° 
w>0 

x° 
x<° 

Brush  Loss. — When  a  condenser  is  operated  in  air  at  high 
voltage,  more  or  less  ionization  occurs  at  the  edges  of  the  plates. 
When  the  edges  of  the  plates  are  entirely  exposed  to  the  air,  as 
in  a  Ley  den  jar,  this  effect  is  large  and  a  considerable  power  loss 
occurs  at  high  voltages.  At  very  high  voltage  the  discharge  is  so 
great  as  to  be  evident  in  the  form  of  a  visible  brush  from  the 
edges  of  the  conducting  plates,  and  is  especially  strong  at  corners 
and  points.  At  such  voltages  the  power  factor  due  to  the  brush 
discharge  is  large  compared  with  that  caused  by  absorption  and 
other  causes. 

Austin  15  has  found  that  the  power  factor  and,  hence,  the 
equivalent  resistance  of  a  Ley  den  jar  in  air  do  not  vary  with 
voltage  up  to  about  10  ooo  volts;  between  10  ooo  and  22  ooo 
volts  they  increase  approximately  as  the  square  of  the  voltage; 
and  at  voltages  greater  than  22  ooo  increase  faster  than  the  square 
of  the  voltage.  Since  the  power  loss  is  proportional  to  the  square 
of  voltage  and  to  the  power  factor,  a  constant  power  factor  means 
power  loss  proportional  to  the  square  of  the  voltage.  Thus,  the 

16  See  reference  No.  air,  Appendix  2. 


Radio  Instruments  and  Measurements  1 29 

power  loss  in  Ley  den  jars  below  10  ooo  volts  is  proportional  to  the 
square  of  the  voltage.  Between  10  ooo  and  22  ooo  volts,  where 
brushing  occurs  and  the  power  factor  is  proportional  to  the  square 
of  voltage,  it  follows  that  the  power  loss  varies  as  the  fourth  power 
of  the  voltage.  Above  22  ooo  volts  the  power  loss  increases  faster 
than  the  fourth  power  of  voltage. 

35.  MEASUREMENT  OF  CAPACITY 

A  capacity  measurement  at  radio  frequencies  is  usually  a  com- 
parison of  the  unknown  capacity  with  a  standard  variable 
condenser.  The  most  precise  measurements  are  made  by  means 
of  a  direct  substitution.  A  primary  standard  for  such  work  is  a 
condenser  which  has  been  calibrated  at  low  frequency  and  is  so 
designed  as  to  have  extremely  low  absorption  and  high  insulation 
resistance,  so  that  its  capacity  is  the  same  at  radio  and  at  low 
frequencies.  For  ordinary  measurements  it  is  possible  to  use  a 
condenser  with  moderate  absorption  and  insulation  resistance 
which  has  been  compared  at  radio  frequencies  with  a  primary 
standard.  A  condenser  with  high  absorption  or  low  insulation 
resistance  may  show  changes  in  capacity  in  the  range  of  radio 
frequencies,  will  increase  the  resistance  of  the  circuit  into  which 
it  is  introduced,  and  is  not  suitable  for  a  standard.  The 
inductance  of  the  leads  within  a  standard  condenser  should  be 
small  in  comparison  with  that  of  the  coil,  or  the  apparent  capacity 
will  vary  with  the  frequency. 

The  unknown  condenser  may  be  fixed  or  variable.  It  is 
inserted  in  series  with  an  inductance  coil  in  a  circuit  which  is 
provided  with  a  device  to  indicate  resonance;  that  is,  a  wave- 
meter  circuit.  The  source  of  oscillations  is  loosely  coupled  to  this 
circuit  and  the  frequency  of  the  source  adjusted  until  resonance 
is  obtained.  The  unknown  condenser  is  then  removed  and  the 
variable  standard  substituted  in  its  place  and  the  setting  of  the 
standard  found  for  which  resonance  is  again  obtained.  The  ca- 
pacity of  the  standard  at  this  setting  is  equal  to  that  of  the 
unknown. 

In  order  to  attain  a  high  accuracy  in  the  comparison,  certain 
precautions  must  be  observed.  The  capacity  of  the  rest  of  the 
circuit  to  each  of  the  two  condensers  must  be  either  very  small 
or  the  same.  On  this  account  it  is  desirable  that  the  leads  to  the 
condenser  be  fairly  long  and  fixed.  Proximity  of  the  metal 
shields,  etc.,  of  the  condensers  to  the  coil  will  reduce  its  inductance 


1 30  Circular  of  the  Bureau  of  Standards 

on  account  of  eddy  currents,  and  since  this  effect  may  vary  with 
the  two  condensers,  it  is  desirable  to  eliminate  it  by  the  use  of 
long  leads.  In  order  that  the  capacities  should  be  definite,  both 
condensers  should  be  shielded,  the  shield  connected  to  one  con- 
denser terminal  and  connected  to  ground,  the  same  lead  from  the 
inductance  coil  being  connected  to  the  earthed  terminal  in  both 
cases. 

When  the  unknown  condenser  is  also  a  variable  and  is  to  be  cali- 
brated at  a  number  of  points,  it  is  convenient  to  use  a  throw-over 
switch  to  change  from  one  condenser  to  the  other.  In  this  case 
it  is  necessary  that  the  inductance  and  capacity  of  the  two  pairs 
of  leads  running  from  the  switch  to  the  condensers  and  mutual 
capacities  to  the  rest  of  the  circuit  should  be  very  nearly  the  same 
in  the  two  positions.  Any  error  from  this  cause  may  be  checked 
by  using  the  same  condenser  first  on  one  set  of  leads  and  then  on 
the  other,  and  comparing  the  settings  for  resonance. 

Calibration  of  Large  Variable  Condenser. — A  large  variable  may 
be  calibrated  against  a  much  smaller  standard  variable  in  the  fol- 
lowing manner.  The  large  unknown  is  first  set  at  a  low  setting 
and  directly  compared  with  the  standard  as  outlined  above.  This 
gives  the  capacity  of  the  large  condenser  at  the  low  setting  with  a 
high  accuracy.  Then  the  two  condensers  are  connected  in  par- 
allel, the  terminals  of  the  two  condensers  which  are  connected  to 
the  shields  being  connected  together.  The  large  condenser  is  set 
at  the  known  point  and  the  small  condenser  at  its  maximum  ca- 
pacity and  the  source  is  adjusted  to  resonance.  Then  keeping  the 
frequency  of  the  source  unchanged,  the  setting  of  the  large  con- 
denser is  increased  by  the  desired  steps,  while  the  setting  of  the 
standard  is  reduced  to  compensate.  From  the  reduction  in  capacity 
of  the  standard,  the  increase  in  capacity  of  the  large  condenser 
from  that  at  the  known  point  can  be  computed.  When  the  stand- 
ard can  be  no  longer  reduced,  the  large  condenser  is  set  at  the 
highest  determined  value,  the  standard  is  again  set  at  its  maximum, 
and  the  wave  length  of  the  source  increased  until  resonance  is  again 
obtained.  The  process  of  increasing  the  capacity  of  the  unknown 
and  decreasing  that  of  the  standard  is  then  repeated. 

Effect  of  Internal  Lead  Inductance. — If  the  unknown  condenser 
has  internal  leads  which  have  appreciable  inductance,  its  apparent 
capacity  will  increase  as  the  wave  length  is  decreased.  To  deter- 
mine this  inductance  the  condenser  is  measured  at  a  very  long  wave 
length  with  a  large  coil  where  the  effect  of  the  leads  will  be  negli- 
gible and  then  with  a  small  coil  of  inductance  L.  Calling  the  ca- 


Radio  Instruments  and  Measurements  131 

pacity  at  long  wave  lengths  C\  and  the  apparent  capacity  at  short 
wave  lengths  C3  and  the  inductance  of  the  leads  /  we  may  write 


or 


C  —C 

\-2        ^1 


This  method  requires  the  use  of  a  standard  condenser  with  leads 

of  negligible  inductance. 

COILS 

36.  CHARACTERISTICS  OF  RADIO  COILS 

To  introduce  a  certain  amount  of  inductance  into  a  circuit,  a  coil 
of  copper  strip  or  specially  stranded  wire  is  ordinarily  used.     This 


fa) 


FIG.  92. — Typical  forms  of  inductance  coils 

is  usually  mounted  or  wound  on  a  form  made  of  such  insulating 
materials  as  wood  or  bakelite  or  other  composition.  The  form  is 
usually  hollow,  and  in  some  cases  the  conductor  is  supported  only  by 
strips  of  insulator  at  intervals.  The  turns  of  wire  or  strip  are  usually 
circular,  but  sometimes  are  polygonal.  Iron  is  not  used  as  a  core 
in  inductance  coils  for  radio  use,  because  such  coils  have  high 
effective  resistance,  the  power  losses  increasing  with  frequency. 
Three  types  of  coil  winding  are  widely  used  in  radio  work ;  the  sin- 
gle layer,  the  flat  spiral  or  pancake,  and  the  multiple  layer.  Cross 
sections  of  these  windings  are  shown  in  (a),  (6),  and  (c)  of  Fig.  92. 


132  Circular  of  the  Bureau  of  Standards 

The  types  (a)  and  (6)  are  universally  used  for  coils  of  low  or  mod- 
erate inductance,  the  type  (6)  is  especially  convenient  for  portable 
instruments  on  account  of  its  compactness.  The  multiple-layer 
coil  is  generally  used  where  it  is  desired  to  obtain  a  large  inductance 
in  a  compact  form. 

The  important  electrical  characteristics  of  a  coil  are  its  induc- 
tance, resistance,  and  capacity.  In  Part  III  are  given  formulas, 
with  tables  and  examples,  which  cover  the  calculation  of  the  in- 
ductance of  practically  all  types  of  coils  which  are  used  in  radio 
circuits. 

On  account  of  the  change  of  current  distribution  within  a  con- 
ductor, the  self -inductance  tends  to  decrease  as  the  frequency  in- 
creases. There  is  no  appreciable  change  of  mutual  inductance 
with  frequency.  The  change  of  self-inductance  is  very  small,  and 
can  be  calculated  only  in  a  few  simple  cases.  The  inductance  of  a 
coil  decreases  somewhat  more  than  the  inductance  of  the  same 
length  of  wire  laid  out  straight.  The  effect  of  distributed  capac- 
ity, however,  is  to  increase  the  inductance.  At  low  frequencies 
the  design  of  coil  form  is  usually  determined  by  the  requirement 
of  minimum  resistance  with  a  given  inductance.  This  is  treated 
on  page  287.  At  radio  frequencies,  however,  the  capacity  of  coils 
is  of  very  great  importance  and  the  choice  of  coil  form  is  largely 
determined  by  the  requirement  of  small  coil  capacity. 

37.  CAPACITY  OF  COILS 

In  section  19  it  has  been  pointed  out  how  the  capacity  of  coils 
may  lead  to  circuits  which  resonate  to  two  wave  lengths  and  how 
overhanging  or  dead  ends  of  coils  may,  on  account  of  the  coil 
capacity,  seriously  affect  the  reactance  and  resistance  of  a  circuit. 
In  this  section  other  cases  will  be  treated  which  show  how  the 
capacity  of  coils  may  considerably  increase  the  resistance  of  radio 
circuits. 

Effect  of  Capacity  on  Inductance  and  Resistance. — In  the  first 
case  it  is  assumed  that  an  emf  (see  Fig.  93(0))  is  introduced  into  a 
circuit  by  means  of  a  coupling  coil  L'  of  few  turns  of  negligible 
inductance  and  not  by  coupling  with  the  main  coil  L.  Under 
these  conditions  the  capacity  of  the  coil  L  will  affect  both  its 
resistance  and  apparent  inductance  and  is  not  merely  added  to 
that  of  the  condenser  as  would  be  the  case  if  the  coupling  coil 
were  removed  and  the  emf  introduced  into  the  circuit  by  induction 
in  the  coil  L  itself.  Instead,  the  coil  L  and  its  capacity  form  a 
•parallel  circuit  as  shown  in  Fig.  93(6). 


Radio  Instruments  and  Measurements 


133 


If  we  write  Z  =  ^R^+uPL^,  where  R&  and  La  are  the  apparent 
resistance  and  inductance  of  the  parallel  circuit,  it  may  be  shown 
that 

\7°) 


>2Co2#2  +  (i-a>2LC0)3 
L(i-co2LC0)-Cy?2 


The  terms  u*-CQ*R2  and  C0RZ  are  very  small  and  negligible,  except- 
ing when  very  close  to  the  frequency  for  which  (i  —  co2LC0)  equals 
zero.  This  is  the  frequency  with  which  the  coil  L  would  oscillate 
by  itself — that  is,  closed  only  by  its  own  capacity.  At  this  fre- 


-  vvw  -  —  vvvvvvr- 

mm 

"c 

c. 

e      _    '. 

FIG.  93. — Equivalent  circuit  of  a  coil  having  distributed 
capacity  placed  in  series  with  a  condenser 

quency  these  terms  determine  the  resistance  and  inductance  of 
the  coil.     For  other  frequencies  we  may  write 


R 


(i-co2LC0)a 
L 

:  (i-o>2LC0) 


(72) 
(73) 


From  these  equations  we  see  that  the  resistance  and  inductance 
start  at  the  values  R  and  L  at  low  frequencies  (co  =  o)  and  increase 
as  the  frequency  increases,  the  resistance  increasing  about  twice 


134  Circular  of  the  Bureau  of  Standards 

as  rapidly  in  per  cent  as  the  inductance.     When  approaching 

the  frequency  co=    ,  -  the  resistance  becomes  very  great  and 
•\LC0 

beyond  this  frequency  falls  off  again,  finally  becoming  zero.  The 
inductance  also  becomes  very  great,  but  just  before  the  frequency 

<a=   i  -  is  attained  we  see  from  equation  (71)  that  it  falls  off 
•\LC0 

very  rapidly  and  becomes  highly  negative.  For  higher  frequen- 
cies it  remains  negative,  but  approaches  zero  as  the  frequency  is 
still  further  increased.  This  negative  value  of  inductance  means 
that  the  coil  is  behaving  as  a  condenser,  and  that  for  the  circuit 
to  resonate  to  such  frequencies  the  condenser  C  would  have  to 
be  replaced  by  an  inductance.  The  primary  interest,  however, 

lies  in  the  frequency  range  below  w  =    ij-~-  for  which  range  it  will 

be  shown  by  means  of  examples  how  the  resistance  and  inductance 
of  L  are  increased. 

Let  us  assume  that  the  coil  L  with  its  leads  has  an  inductance 
of  a  millihenry  (io~3  henry)  and  a  capacity  of  50  micromicrofarads 
(5X  io~n  farad). 
If  <o  =  io6,  then  (i  -«2LC0)  =  (i  -  io12X  io~3  X  5  X  io~u)  =0.95 

T-> 

and  R&  =  ,   "   ^  =  i  .  1  1  R,  an  increase  of  1  1  per  cent. 

(o.95)2 

=  1.05  L,  an  increase  of  5  per  cent. 


&     -(  -  r 

The  wave  length  corresponding  to  this  frequency  is  A  =  1885 
meters,  and  the  capacity  C  of  the  condenser  for  resonance  is 
950  micromicrofarads. 

If  (o  =  1.5  X  io6  corresponding  to  a  wave  length  of  1250  meters, 
the  resistance  increase  is  26  per  cent  and  the  inductance  increase 
12  per  cent.  The  capacity  C  would  then  be  400  micromicrofarads. 

These  are  examples  of  conditions  that  may  readily  occur  in 
practice.  It  is  very  desirable  in  designing  radio  circuits  to 
arrange  whenever  possible  that  the  emf  be  introduced  in  the 
main  coil  L  itself. 

Effect  of  Dielectric  Absorption  in  Coil  Capacity.  —  Even  in  the  case 
of  a  simple  oscillatory  circuit  of  coil  and  condenser  with  the  emf 
applied  by  induction  in  the  coil  (in  which  case  the  coil  capacity 
may  be  considered  as  in  parallel  with  that  of  the  condenser)  the 
resistance  of  the  circuit  may  be  increased.  This  is  not  due  merely 
to  the  capacity  of  the  coil,  but  because  the  coil  capacity  is,  in 


Radio  Instruments  and  Measurements  135 

general,  a  highly  absorbing  condenser.  The  dielectric  is  the 
insulation  of  the  wire  and  the  material  of  the  coil  form,  and  the 
resulting  phase  difference  of  the  coil  capacity  may  be  several 
degrees.  The  resistance  added  to  the  circuit  on  this  account 
will  be  larger  the  greater  the  absorption  of  the  coil  capacity, 
and  the  larger  this  capacity  relative  to  the  condenser  capacity, 
and  also  the  longer  the  wave  length. 

To  illustrate  the  importance  of  this  effect,  let  us  take  the  same 
data  as  used  in  the  preceding  examples,  with  the  further  assump- 
tion that  the  coil  capacity  C0  has  a  phase  difference  of  2°. 

In  the  first  example  co  =  io6,  C0  =  5O  micromicrofarads,  and 
C  =  95<D  micromicrofarads.  The  phase  difference  of  C0  and  C  in 

parallel  will  be  -  -  times    2°  =  6'.     The  effective  resistance 

950  +  50 

added  to  the  circuit  on  account  of  absorption  will  be 

• 

tan  6          tan  6' 


co(C  +  C0)      io6(io~9) 


1.75  ohms. 


In  the  second  example,  w  =  1.5X10°,  C0  =  50  micromicrofarads, 
C  =  400  micromicrofarads.  Hence,  6  =  1 3 .  '3 

tan  13/3  3.88  Xio-3 

p  =  1.5  X  io«(o.45  X  io-)  =0.67  x  10-3  =  5-8  ohms. 

It  is  therefore  clear  that  it  is  extremely  important  so  to  design 
coils  that  the  capacity  shall  be  small  and  of  as  low  absorption  as 
possible. 

Reduction  of  Coil  Capacity. — The  coils  of  type  (a)  and  (6)  have 
low  capacities,  while  type  (c)  has  a  large  capacity.  This  results 
from  the  fact  that  in  a  coil  of  type  (a)  or  (6)  with  n  turns,  only 

-  th  of  the  total  voltage  acting  upon  the  coil  is  impressed  between 

/ib 

the  adjacent  turns,  while  in  the  multiple-layer  coil  as  ordinarily 
wound,  turns  which  have  high  voltages  between  them  will  be 
adjacent.  For  example,  in  a  coil  of  two  layers,  if  one  layer  is 
first  wound,  and  then  the  other  on  top  of  it,  the  first  and  last 
turns  of  the  coil  will  be  adjacent  and  the  total  voltage  on  the  coil 
will  be  impressed  between  them.  The  capacity  of  such  a  coil 
introduces  serious  errors  in  high-frequency  measurements  unless 
the  coil  is  accurately  standardized. 

The  capacity  of  coils  of  type  (a)  or  (6)  can  be  reduced  and  the 
phase  difference  improved  if  the  windings  are  separated  slightly 


136 


Circular  of  the  Bureau  of  Standards 


with  air  between.  On  account  of  the  high  voltage  between  the 
adjacent  turns  this  is  a  customary  procedure  in  the  case  of  coils 
designed  to  carry  large  currents,  such  as  the  familiar  flat  spiral 
coils  used  in  transmitters.  In  the  standard  wave-meter  coils  of 
this  Bureau  this  method  of  reducing  the  capacity  and  phase  differ- 
ence has  been  used.  The  coils  are  shown  in  Fig.  215,  facing  page 
318,  and  range  in  inductance  from  60  to  about  5000  microhenries. 
The  coil  capacities,  including  fairly  long  leads,  range  from  9  to  16 
micromicrofarads.  No  data  are  available  as  to  the  phase  differ- 
ences due  to  these  coil  capacities,  but  since  the  windings  are 
separated  from  each  other  with  air  between,  excepting  at  the 
points  of  support  on  the  coil  form,  the  phase  differences  are 
extremely  low. 

In  coils  of  type  (c),  Fig.  92,  the  capacity  may  be  reduced  by 
using  the  so-called  "banked"  winding.  Instead  of  winding  one 
layer  complete  and  then  winding  the  next  layer  back  over  the  first, 


FIG.  94. — Method  of  banking  tJie  -winding  of  a  coil 
in  order  to  reduce  its  distributed  capacity 

one  turn  is  wound  successively  in  each  of  the  layers,  the  winding 
proceeding  from  one  end  of  the  coil  to  the  other.  The  best  results 
are  obtained  in  a  coil  of  a  few  layers.  The  method  is  illustrated 
in  Fig.  94  for  a  coil  of  two  layers,  the  succession  of  the  turns  being 
indicated  by  the  numbers.  The  turns  in  the  lower  layer  are  pre- 
vented from  slipping  during  the  winding  by  grooving  the  coil  form 
or  covering  it  with  rubber  tape.  The  maximum  voltage  between 
adjacent  wires  in  this  two-layer  coil  is  the  voltage  corresponding 
to  three  turns. 

38.  MEASUREMENT  OF  INDUCTANCE  AND  CAPACITY  CF  COILS 

When  the  inductance  of  a  coil  is  measured  at  high  frequencies, 
the  quantity  obtained  is  the  apparent  inductance;  that  is,  a 
quantity  in  which  is  combined  the  effects  of  the  pure  inductance 
and  the  capacity  of  the  coil.  (See  p.  132.)  Thus  it  is  customary 
to  speak  of  the  inductance  of  a  coil  at  a  given  wave  length  or 
frequency.  The  inductance  of  a  coil  at  a  given  frequency  is 


Radio  Instruments  and  Measurements  137 

measured  by  comparing  it  with  a  standard  coil  or  by  determining 
the  capacity  required  in  parallel  with  the  coil  to  form  a  circuit 
which  is  resonant  to  the  known  frequency. 

In  the  first  method  let  L8  be  the  inductance  of  the  standard  coil 
and  Lx  that  of  the  unknown.  The  source  is  set  at  a  given  fre- 
quency and  a  circuit  containing  L8  and  a  condenser  is  tuned  to 
resonance  with  the  source  and  the  condenser  setting  read.  Let 
the  capacity  be  C,.  Then  Lx  is  substituted  and  resonance  again 
obtained,  the  capacity  being  Cx.  Since  the  frequency  is  the  same 
in  both  cases  we  have 

J~*Q    *-*  B  *~~  -L^x    *-*  Z 

Ll^B  Ca 
I=~~ 


Here  L.  is  the  apparent  inductance  of  the  standard  coil  at  the 
frequency  of  the  measurement.  If  the  pure  inductance  L9  and 
capacity  C0  of  the  standard  coil  is  given 

Lp  (Ce  +  C0)  =  Lx  Cx 

T         —       P   ^     8       ^-o/ 

~~cT 

In  the  second  method,  the  frequency  or  wave  length  furnished 
by  the  source  must  be  accurately  known.  The  circuit  with  the 
unknown  coil  Lx  is  adjusted  to  resonance  with  the  source.  If  the 
resonant  capacity  is  Cx  we  have 


r       3-553  *2 

*-»••     r~ 

t-x 

where  X  is  wave  length  in  meters. 

Cx  is  capacity  in  micromicrofarads. 

LK  is  inductance  in  microhenries. 

Determination  of  Coil  Capacity.  —  The  pure  inductance  and 
capacity  of  the  coil  may  be  derived  from  the  observed  values  of 
apparent  inductance  in  various  ways.  A  simple  method  makes 
use  of  the  formula  given  on  page  133  for  the  apparent  inductance  in 
terms  of  the  pure  inductance  and  coil  capacity.  Denoting  two 
frequencies  or  capacity  settings  by  subscripts  i  and  2  we  have 


-(•4:) 


JL/a.9        • 


138 
or 


Circular  of  the  Bureau  of  Standards 


Co 


or  approximately 


and 


,        o 

r 

*- 


r     r 

,    *~o «--o - 

i  -f- ^  ~7r~ 

*-*  1       *-  2       - 


—i    ai     T  I_CL~IL_ 
~\ /    ~   /r  — r 

\-^a2          /L2       *-l 


Knowing  C0,  L  may  be  obtained  by  substitution. 

If  one  determination  is  made  with  a  very  large  condenser 
capacity  and  another  with  a  small  condenser,  we  may  write 

Lai  =  L  approx. 
La2=L(I+CV 


and 


-G0  °  G 

FlG.  95. — Graphical  method  of  determining  the  capac- 
ity of  a  coil 


fe-) 


Another  method  which  is  likewise  of  considerable  value  in  wave- 
meter  calibrations  makes  use  of  a  plot  of  the  wave  length  squared 
against  the  condenser  capacity.  Since 


x  =  1.885 

X2  =  3-553  MC  +  C0) 


Radio  Instruments  and  Measurements  139 

Since  L,  the  pure  inductance,  is  a  constant,  the  relation  between  X2 
and  C  is  linear  and  the  plot  will  be  a  straight  line  as  in  Fig.  95. 

7    /"\  2 

The  slope  of  this  line  ,~  =3-553  L  determines  the  pure  induc- 
tance of  the  coil.  The  distance  from  the  origin  to  the  intercepts 
of  the  line  with  the  axis  of  abscissae  determines  the  capacity  of  the 
coil,  for  when  X2  =o;  C  —  —  C0.  Another  method  of  determining 
the  coil  capacity  making  use  of  the  harmonic  oscillations  of  a 
pliotron  generator  has  already  been  described  in  the  section  on 
wavemeters,  page  100. 

CURRENT  MEASUREMENT 
39.  PRINCIPLES 

Current  measurements  at  radio  frequencies  are  made  in  trans- 
mitting sets,  in  wave  meter  and  other  testing  circuits,  and  in 
receiving  sets.  The  instruments  used  in  such  measurements,  their 
principles,  and  the  precautions  necessary  in  using  them,  are  dis- 
cussed in  this  section.  The  measurement  of  current  is  a  cardinal 
operation  in  high-frequency  work,  to  a  much  greater  degree  than 
at  low  frequencies,  since  upon  it  depends  also  the  measurement  of 
resistance,  and  it  is  involved  in  most  of  the  measurements  of  other 
quantities.  There  are  three  distinct  ranges  of  currents:  Large 
currents  such  as  are  used  in  large  transmitting  sets ;  moderate  cur- 
rents in  small  sending  outfits  and  in  wave  meters,  decremeters,  etc., 
used  at  a  sending  station;  and  the  very  small  currents  in  receiving 
circuits  and  in  testing  equipments  operated  by  buzzers  and  other 
weak  sources.  These  three  ranges  will  be  treated  separately, 
beginning  with  the  middle  range  of  moderate  currents,  because  the 
simplest  instruments  are  there  used  and  the  fundamental  prin- 
ciples can  best  be  discussed  in  connection  with  them. 

Most  of  the  ammeters  used  for  low-frequency  currents  are 
entirely  unsuitable  at  high  frequency.  Most  of  them  are  ruled 
out  by  the  requirement 16  that  the  circuit  within  a  high- 
frequency  ammeter  must  be  of  the  greatest  simplicity,  because  its 
inductance  and  its  capacity  must  be  as  small  as  possible.  Large 
impedance  and  large  capacity  would  tend  to  make  the  current 
flow  through  the  dielectric  as  well  as  through  the  conductor,  in 
amount  varying  with  the  frequency.  This  would  cause  the  read- 
ings of  the  instrument  to  change  with  frequency.  The  require- 

16  See  reference  No.  195,  Appendix  2. 


140  Circular  of  the  Bureau  of  Standards 

ment  of  a  simple  form  of  circuit  is  best  fulfilled  by  a  single  straight 
wire  of  very  small  diameter.  The  current  flowing  in  such  a  con- 
ductor is  most  simply  measured  by  its  heating  effect,  and  hence 
it  has  come  about  that  most  ammeters  for  high  frequency  have 
been  of  the  hot-wire  or  thermal  type.  Other  types  have  been 
used  to  a  limited  extent. 

The  Thermal  Ammeter.  —  The  basic   principle  of  the  thermal 
ammeter  is  given  by 


P  being  the  power  consumed  as  heat  in  the  instrument.  The 
deflection  of  the  instrument  depends  upon  the  heating  effect  P, 
which  is  indicated  in  one  of  the  various  ways  discussed  below. 
In  order  that  a  given  deflection  should  always  correspond  to  the 
same  current  it  is  necessary  that  the  relation  of  P  to  I  should 
remain  constant,  and  this  requires  that  R,  the  resistance  of  the 
instrument,  should  not  change  with  frequency.  In  order  not  to 
vary  in  resistance  at  radio  frequencies,  a  conductor  must  have 
very  small  cross  section  (see  section  below  on  Resistance)  .  Hence, 
the  working  element  of  a  thermal  ammeter  must  be  a  fine  wire  or 
a  very  thin  strip  of  metal.  For  the  accuracy  required  in  most 
radio  work  it  is  only  necessary  that  the  resistance  of  the  wire  or 
strip  should  not  change  by  more  than  i  per  cent  over  any  range 
of  frequency  used.  The  largest  wire  which  may  be  used  in  an 
ammeter  is  thus  found  by  consulting  Table  18,  page  310,  for  the 
highest  frequency  to  be  used.  In  ammeters  which  are  intended 
for  use  at  the  highest  frequencies  of  radio  practice,  viz,  about 
2  ooo  ooo  (corresponding  to  a  wave  length  of  150  meters),  the 
largest  diameter  permissible  for  a  copper  wire  is  0.08  mm  and  for 
a  constantan  wire  is  0.4  mm.  Using  a  single  wire  of  such  a  size, 
only  a  few  amperes  can  be  measured,  because  larger  currents 
would  overheat  the  wire  and  alter  its  properties. 

The  length  of  the  wire  is  not  important;  it  must  simply  be  long 
enough  so  that  the  current  distribution  within  it  is  not  appre- 
ciably altered  by  the  terminals  to  which  it  is  connected.  A  length 
of  10  cm  or  more  is  satisfactory.  If  properly  constructed,  such 
instruments  may  be  calibrated  with  direct  current  or  low- 
frequency  alternating  current  and  the  calibration  assumed  correct 
at  high  frequency.  It  is  always  safest,  however,  to  calibrate  the 
ammeter  by  comparison  with  some  recognized  standard  at  the 
frequencies  at  which  it  will  be  used. 


Radio  Instruments  and  Measurements  141 

Sources  of  Error. — Certain  precautions  must  be  observed  in  the 
use  of  high-frequency  ammeters;  some  of  these  will  be  pointed 
out  below  in  connection  with  particular  types  of  instruments. 
There  is  one  source  of  error  to  which  all  are  subject,  whether 
thermal  ammeters  or  not,  at  extremely  high  frequencies.  For 
very  short  wave  lengths,  the  current  is  usually  not  the  same 
at  all  points  of  a  circuit.  The  capacity  between  parts  of  the 
circuit  and  the  surroundings  is  so  important  that  an  appreciable 
fraction  of  the  current  is  shunted  through  the  dielectric  and  so 
different  amounts  of  current  flow  in  different  parts  of  the  wire 
circuit.  At  such  frequencies  one  must  be  careful  to  place  the  am- 
meter at  that  point  of  the  circuit  where  the  value  of  the  current  is 
desired.  In  order  to  diminish  the  flow  of  current  away  from  the 
circuit,  special  devices  are  used  in  some  ammeters  to  prevent 
grounding  of  the  circuit  to  the  ammeter  case.  For  example, 
where  the  deflection  of  the  instrument  depends  on  thermal  ex- 
pansion a  short  length  of  insulation  may  be  placed  in  the  indi- 
cating wire;  and  in  instruments  depending  on  the  heating  in  a 
thermocouple,  the  heater  and  the  thermocouple  may  be  separated 
instead  of  in  contact. 

40.  AMMETERS  FOR  SMALL  AND  MODERATE  CURRENTS 

The  ammeters  used  for  measuring  radio  currents  of  about  0.003 
to  3  amperes  are  hot-wire  instruments  of  the  simplest  form. 
They  consist  essentially  of  a  single  fine  wire,  with  a  means  of 
indicating  the  heat  produced.  The  heat  production  may  be  indi- 
cated by  any  thermometric  method,  and  the  following  are  in  use: 
Expansion,  thermoelectric  effect,  and  calorimetry.  These  three 
types  of  instrument  are  discussed  in  the  following. 

Current-Square  Meter. — The  most  familiar  instrument  of  the 
expansion  type  is  the  current-square  meter,  used  in  wave  meters 
and  decremeters.  This  has  sometimes  been  called  a  "watt- 
meter, "  the  deflections  being  proportional  to  the  watts  consumed 
within  the  instrument  itself;  such  a  name  is  utterly  misleading 
and  undesirable.  The  hot  wire  is  of  resistance  metal,  and  its 
increase  in  length  when  heated  is  indicated  by  a  pointer  operated 
by  a  thread  attached  to  the  wire  and  held  taut  by  a  spring.  The 
scale  usually  has  graduations  equally  spaced,  so  that  the  readings 
are  proportional  to  the  square  of  the  current.  This  makes  the 
instrument  particularly  useful  in  measurements  of  resistance  by 
the  reactance  variation  method.  The  scale  can,  of  course,  be 


1 42  Circular  of  the  Bureau  of  Standards 

graduated  so  as  to  read  current  directly,  but  such  a  scale  is  badly 
crowded  at  the  lower  end. 

A  familiar  type  of  this  instrument  gives  full-scale  deflection 
with  0.08  ampere.  The  instrument  is  shown  in  Fig.  96,  facing 
page  156.  It  has  a  resistance  of  about  5  ohms.  If  intended  for 
use  in  actual  measurements  of  current  the  instrument  can  not 
be  shunted,  for  reasons  explained  in  the  next  section.  For  use 
in  a  wave  meter  or  decremeter,  however,  it  may  be  shunted  to 
give  whatever  current  range  may  be  desired,  since  in  such  use 
the  variations  of  frequency  are  so  small  during  any  one  measure- 
ment that  the  shunt  does  not  affect  the  accuracy  of  the  result. 

A  source  of  serious  error  in  the  use  of  these  instruments  is  the 
presence  of  electrostatic  charges  on  the  glass  covering  the  dial. 
If  the  glass  happens  to  be  stroked  with  any  object  the  pointer 
may  deflect  several  scale  divisions  and  remain  in  the  new  posi- 
tion a  long  time.  This  is  not  an  error  which  can  be  eliminated 


Hotwire 


Cu     |  F»         I «  Const. 


FIG.  97. — Simple  hot-wire  ammeter  with  thermoelectric  indi- 
cating device 

by  using  the  zero  adjustment,  because  the  electrostatic  charges 
on  the  glass  exert  a  force  on  the  pointer  which  varies  with  the 
position  of  the  pointer.  The  most  convenient  way  to  eliminate 
this  effect  is  to  breathe  on  the  glass,  the  charge  being  conducted 
away  by  the  layer  of  moisture.  This  effect  is  especially  trouble- 
some in  cold  weather,  when  electrostatic  charges  are  readily 
produced  and  maintained. 

All  hot-wire  instruments  show  more  or  less  zero  shift.  After 
current  has  been  flowing  through  the  instrument,  the  pointer 
does  not  return  exactly  to  its  original  position.  In  a  current- 
square  meter  having  a  zero  adjustment  this  is  usually  overcome 
by  first  allowing  current  to  flow  for  several  seconds  without  taking 
a  reading,  then  cutting  off  current  and  setting  zero,  and  then 
allowing  current  to  flow  and  reading.  If  the  instrument  has  no 
zero  adjustment,  error  may  be  eliminated  by  first  allowing  cur- 


Radio  Instruments  and  Measurements  143 

rent  to  flow  without  taking  a  reading,  then  cutting  it  off  and 
reading  zero,  then  reading  current,  and  then  reading  zero  again. 
The  mean  of  the  two  zero  readings  is  to  be  subtracted  from  the 
current  reading.  (In  instruments  having  a  nonuniform  scale  the 
zero  reading  should  be  made  as  explained  on  p.  149.) 

Thermoelectric  Ammeter. — The  heat  developed  in  the  hot  wire 
may  be  indicated  by  means  of  a  thermocouple,  placed  very  near 
or  in  contact  with  the  wire.  The  electromotive  force  produced 
by  the  heating  of  the  thermocouple  is  measured  by  a  suitable 
direct-current  instrument.  The  indications  depend  upon  the 
temperature  at  one  point  only  of  the  hot  wire  instead  of  upon 
the  heating  effect  throughout  the  whole  wire  as  in  the  expansion 
ammeter.  There  are  two  types  in  use  for  moderate  currents.  A 
diagram  of  the  simple  type  is  shown  in  Fig.  97.  A  is  the  fine 
wire  which  carries  the  high-frequency  current. 
The  copper-constantan  thermocouple  is  hard 
soldered  to  A  and  connected  to  the  binding  posts 
and  thence  to  a  galvanometer.  Such  an  instru- 
ment is  easily  constructed  for  laboratory  use  for 
currents  up  to  2  amperes.  Commercial  instru- 
ments are  made  by  combining  the  hot  wire, 
thermocouple,  and  a  pointer-type  microammeter 
into  a  single  instrument. 

The  other  type  of  thermoelectric  ammeter  is 
the  crossed-wire  type,  illustrated  in  Fig.  98.  FIG.  98.— Crossed- 
Two  fine  wires,  Ab  and  aB,  of  constantan  and 
copper,  constantan  and  platinum,  or  similar  pair 
of  metals,  are  in  contact  at  one  point,  crossing  each  other  as  shown. 
Connection  is  made  from  the  points  A  and  B  to  the  source  of  high- 
frequency  current.  The  current  heats  the  fine  wires,  raising  the 
temperature  of  the  junction,  and  this  causes  an  emf  between  a 
and  b,  to  which  points  a  microammeter  or  galvanometer  is  con- 
nected. The  deflections  of  the  usual  thermoelectric  ammeters  are 
approximately  proportional  to  the  square  of  the  measured  current. 

Thermoelectric  ammeters  should  be  calibrated  with  low-fre- 
qency  alternating  rather  than  direct  current,  for  the  reasons 
stated  on  page  170,  below. 

Air-Thermometer  Ammeter. — This  instrument,  which  is  really  a 
calorimeter,  was  formerly  used  in  wave  meters  and  in  measure- 
ments of  high-frequency  resistance,  for  currents  of  a  few  hun- 
dredths  to  a  few  tenths  ampere.  Current  passing  through  the 

35601°— 18 10 


144 


Circular  of  the  Bureau  of  Standards 


fine  wire  (Fig.  99)  heats  the  air  in  the  glass  bulb  and  causes  the 
alcoholin  the  right  side  of  the  U-tube  to  rise.  In  order  to  elim- 
inate the  effect  of  other  heat  than  that  produced  in  the  hot  wire 
it  is  desirable  to  inclose  the  bulb  in  a  vacuum  jacket.  A  null 
instrument  is  easily  made  on  the  air- thermometer  principle,  by 
connecting  a  bulb  to  the  right  side  of  the  U-tube,  entirely  similar 
to  the  one  shown  on  the  left  side.  If  the  two  sides  are  exactly 


FIG.  99.  —  Air-thermometer  ammeter 

alike,  the  liquid  column  will  not  move  when  the  RP  in  one  wire 
equals  the  RP  in  the  other.  These  instruments  are  no  longer  used 
in  practice. 

41.  THERMAL  AMMETERS  FOR  LARGE  CURRENTS 

It  is  a  common  dictum  that  ammeters  used  in  measuring  alter- 
nating currents  of  even  rather  low  frequency  must  not  be  shunted. 
This  is  obvious  from  consideration  of  the  expression  relating 
the  current  7t  in  the  shunt  and  /2  in  the  instrument,  neglecting 
mutual  inductance, 


since  usually  the  inductances  of  the  instrument  and  the  shunt  are 
in  a  different  ratio  from  that  of  the  resistances,  and  hence  the 
distribution  of  the  current  varies  with  frequency.  In  fact,  even 

L     R  1  2 

if  -^  =  —2,  which  would  make  the  expression  for  -^  independent 
^     KI  12 

of  co,  the  current  distribution  in  an  actual  case  is  likely  to  vary 


Radio  Instruments  and  Measurements  145 

with  frequency  "because  of  the  mutual  inductance.  Now,  any  high- 
frequency  ammeter  in  which  the  circuit  within  the  ammeter  itself 
consists  of  more  than  a  single  elementary  filament — or  its  closest 
approximation,  a  fine  wire — in  reality  involves  shunting,  and 
needs  most  careful  consideration  before  it  can  be  pronounced  free 
from  error.  The  proper  design  of  ammeters  suitable  for  meas- 
uring radio  currents  up  to  several  hundred  amperes  is  therefore 
a  difficult  matter. 

Parallel  Wires  or  Strips. — When  larger  currents  than  about  3 
amperes  have  to  be  measured,  the  single  wire  will  not  suffice; 
since  the  wire  must  be  of  so  small  a  diameter  that  its  resist- 
ance is  not  changed  by  frequency,  large  currents  will  overheat 
it.  To  measure  large  high-frequency  currents,  therefore,  the  cur- 
rent must  have  more  than  one  path,  and  it  is  common  to  use 
either  wires  or  very  thin  strips  of  metal  in  parallel.  This  amounts 
to  shunting,  and  it  is  therefore  difficult  to  make  these  ammeters 
accurate  at  radio  frequencies.  Many  of  those  in  use  have  large 
errors.  The  heating  effect  is  indicated  either  by  an  expan- 
sion device  or  by  a  thermocouple  attached  to  one  of  the  wires 
or  strips.  If  the  ratio  of  the  current  in  that  particular  wire  or 
strip  to  the  total  current  changes  with  frequency,  the  instrument 
will  be  proportionately  in  error.  The  errors  are  much  larger 
than  they  would  be  if  the  indicated  current  depended  on  the  heat 
production  of  the  whole  current,  since  the  changes  of  current  dis- 
tribution within  a  particular  system  are  changes  of  the  first  order 
of  magnitude  compared  to  which  the  change  of  heat  production 
in  the  whole  system  is  of  the  second  order.17 

The  simplest  arrangement  of  parallel  elements  which  has  been 
used  in  this  class  of  ammeters  is  a  group  of  fine  wires  or  thin 
strips  all  in  a  single  plane,  as  in  Fig.  100.  The  arrangements 
shown  in  these  diagrams  are  subject  to  several  errors.  In  the 
first  place,  the  self -inductances  of  the  lugs  to  which  the  wires  or 
strips  are  fastened,  wrhile  utterly  negligible  at  low  frequencies, 
contribute  a  large  part  of  the  impedance  at  high  frequencies. 
Because  of  this,  the  current  in  the  wire  z  (Fig.  100)  will  decrease 
relative  to  the  current  in  the  wire  u  as  the  frequency  increases. 
This  source  of  error  may  be  eliminated  by  connecting  the  current 
leads  at  opposite  corners,  instead  of  the  two  adjacent  corners  as 
shown. 

17  See  reference  No.  195,  Appendix  2. 


146 


Circular  of  the  Bureau  of  Standards 


Effect  of  Mutual  Inductances. — A  more  important  source  of 
error  is  the  mutual  inductance  between  the  wires  or  strips.  The 
mutual  inductance  between  x  and  z  is  less  than  the  mutual  induct- 
ance between  x  and  y,  Fig.  100.  The  combined  effect  of  all  the  mu- 
tual inductances  is  that  the  reactance  of  the  outer  wires  is  less  than 
of  the  inner  wires.  The  resistances  of  the  wires  determine  the 
distribution  of  current  at  low  frequencies,  but  as  the  frequency 
is  increased  the  reactances  become  more  and  more  important. 
Consequently,  at  high  frequencies  more  current  flows  in  the  outer 
wires  than  in  those  near  the  center.  This  effect  is  greater  the  closer 
the  wires  or  strips  are  spaced.  If  the  indicating  device  (expan- 
sion arrangement  or  thermocouple)  is  placed  on  one  of  the  out- 
side wires  or  strips,  the  ammeter  reads  high  at  high  frequencies, 


B' 


FlG.  100. — Ammeters  with  heating  elements  in  parallel 

and  if  on  one  near  the  center  of  the  group,  it  reads  low  at  high 
frequencies. 

The  errors  due  to  mutual  and  self  inductance  can  be  avoided 
by  arranging  the  wires  or  strips  as  equidistant  elements  of  a 
cylinder  and  leading  the  current  in  to  the  centers  of  the  ends  of 
the  cylinder.  The  self -inductance  is  the  same  in  each  current 
path,  and  each  has  the  same  set  of  mutual  inductances  with 
respect  to  the  others.  The  currents  in  the  different  paths  must 
then  be  the  same  at  high  frequencies.  It  does  not  follow  that 
such  an  instrument  is  free  from  error,  for  the  reason  that  the 
resistances  of  the  individual  wires  or  strips  may  be  quite  different 
because  of  variations  of  hardness  and  small  variations  of  cross 
section.  It  is  very  difficult  in  practice  to  get  such  thin  wires  or 
strips  of  uniform  thickness.  The  inductances  do  not  vary  appre- 
ciably with  variation  of  cross  section  while  the  resistances  do  of 


Radio  Instruments  and  Measurements  147 

course,  differ  on  this  account.  Thus  the  current  distribution 
among  the  several  current  paths  is  not  uniform  at  low  frequen- 
cies, where  resistance  is  the  determining  factor,  while  it  is  uniform 


FIG.  101. — Ammeter  with  the  parallel  heating  elements  arranged  cylindrically 

at  frequencies  so  high  that  only  the  inductances  determine  it.  Am- 
meters of  this  type  are  used  for  measuring  currents  up  to  300 
amperes.  A  commercial  ammeter  of  this  type  is  shown  in  Fig. 
102,  facing  page  156.  They  are,  in  general,  satisfactory,  but  the 
readings  change  with  frequency  by  a  few  per  cent  for  the  reason 
just  explained.  The  residual  errors  in  the  cylindrical  type  of 
ammeter  may  be  practically  eliminated  by  the  use  of  a  thermo- 
couple on  each  wire  or  strip. 

Advantage  of  High-Resistance  Elements. — It  is  an  interesting 
fact  that  the  changes  in  current  distribution  in  thermal  ammeters 
occur  just  in  the  range  of  radio  frequencies.  This  is  the  reason 
why  it  is  difficult  to  design  such  instruments  properly.  The  size 
of  the  conductors  used  (wires  having  diameters  of  the  order  of 
o.i  mm  and  strips  of  that  thickness)  is  such  that  their  inductive 
reactance  is  negligible  compared  to  their  resistance  below  a  fre- 
quency of  about  100  ooo  but  is  much  greater  than  the  resist- 
ance above  2  ooo  ooo,  so  that  the  current  distribution  depends 
on  very  different  properties  outside  these  limits  and  passes  through 
a  variation  within  this  range.  The  greater  the  resistance  of  the 
conductors  the  higher  are  these  limiting  frequencies.  By  using 
high-resistivity  wires  or  strips,  therefore,  the  errors  of  these 
instruments  can  be  moved  up  to  rarely  used  frequencies.  There 
is  a  limit  to  this  use  of  high  resistance,  in  that  the  increase  of 
resistance  means  that  the  conductor  gets  hotter,  or  can  carry  less 
current  for  a  given  temperature  rise.  Platinum-iridium  or  plati- 
num-rhodium are  desirable  materials  in  view  of  the  requirements, 
having  moderately  high  resistivity  and  being  capable  of  standing 
high  temperatures.  With  very  fine  wires  or  very  thin  strips  of 
these  materials,  well  separated  but  not  necessarily  arranged  in 
the  cylindrical  form,  ammeters  for  currents  up  to  20  amperes  are 
reliable  for  all  except  the  highest  radio  frequencies. 


148  Circular  of  the  Bureau  of  Standards 

So-Called  Unshunted  Ammeter. — A  type  of  thermal  ammeter 
which  has  been  used  extensively  to  measure  radio  currents  up  to 
10  amperes  is  mentioned  here  only  to  point  out  its  utter  unsuita- 
bility.  This  is  the  so-called  unshunted  ammeter,  represented 
diagrammatically  in  Fig.  103.  A  and  B  are  the  current  leads. 
They  connect  to  thick  copper  bars,  from  which  flexible  silver  strips 
take  the  current  to  several  points  of  the  hot  wire,  whose  expan- 
sion is  measured  by  the  ordinary  device  (not  shown  in  Fig. 
103).  Thus  a  single  wire  carries  the  whole  current,  and  the  in- 
strument is  called  unshunted.  The  resistances  of  the  copper 
bar  and  metal  strips  are  negligible  in  comparison  with  the  resist- 
ance of  the  hot  wire,  and  if  the  lengths  of  the  sections  in  parallel 
are  the  same,  each  carries  the  same  current  as  any  other  on  low 


L 


COPPER   BAR 


HOT  WIRE 


f  FLEXIBLE 
i  STRIP 


COPPER  BAR 


FIG.  103. — So-called  unshunted  ammeter  of  four  sections 

frequency.  But  the  inductances  of  these  parts  are  by  no  means 
negligible,  and  consequently  on  high  frequency  the  different 
portions  of  the  wire  carry  different  amounts  of  current.  In  fact, 
in  practical  cases  the  impedance  of  the  hot  wire  itself  is  but  a 
small  part  of  the  impedance  of  each  current  path  for  high  fre- 
quencies. Even  the  mutual  inductances  of  the  different  por- 
tions of  the  "hot  wire"  are  not  negligible  and  in  themselves 
tend  to  cause  more  current  to  flow  in  the  central  sections  than 
in  the  outer  sections.  The  changes  in  current  distribution  from 
the  uniformity  of  direct-current  distribution  are,  in  fact,  very 
large.  They  are  equivalent  to  an  increase  in  the  resistance  of 
the  system  as  a  whole,  so  that  these  instruments  read  high  on 


Radio  Instruments  and  Measurements  149 

high  frequency.  Whenever  an  accuracy  of  10  per  cent  or  better 
is  desired,  this  type  of  ammeter  should  not  be  used. 

Errors. — There  are  a  few  special  precautions  to  be  observed  in 
the  use  of  thermal  ammeters  for  large  high-frequency  currents. 
The  leads  should  be  brought  straight  in  to  the  instrument;  if  run 
alongside  the  instrument  so  as  to  be  parallel  to  its  working  parts, 
the  current  in  the  lead  will  have  a  greater  mutual  inductance  with 
some  of  the  current  paths  than  others  and  will  cause  a  change  in 
current  distribution  which  may  disturb  the  reading.  In  using 
any  ammeter  of  the  hot-strip  type  cafe  must  be  taken  as  to  the 
position  of  the  instrument.  The  readings  vary  with  position 
because  of  altered  heat  convection  from  the  strip.  This  effect  is 
approximately  the  same  at  low  and  high  frequencies.  Error  is 
avoided  by  always  using  the  instrument  in  the  position  in  which 
it  was  calibrated.  The  difference  between  the  readings  with  the 
instrument  in  such  positions  that  the  strip  is  horizontal  and  strip 
vertical  has  been  found  to  be  of  the  order  of  10  per  cent. 

All  thermal  instruments  show  a  zero  shift,  which  is  troublesome 
in  accurate  measurements.  In  the  usual  ammeters  it  may  amount 
to  several  per  cent  of  full-scale  deflection.  To  make  a  measure- 
ment of  any  accuracy  it  is  necessary  to  leave  the  current  on  for 
at  least  several  seconds,  because  the  wire  does  not  heat  up  instan- 
taneously and  the  pointer  does  not  come  immediately  to  its  final 
position.  When  the  current  is  then  cut  off,  the  pointer  does  not 
return  exactly  to  its  original  position.  The  difficulty  may  be 
overcome  when  the  instrument  has  a  zero  adjustment  by  the  fol- 
lowing procedure:  Allow  the  current  to  flow  for  several  seconds 
without  making  a  reading,  cut  it  off,  and  after  a  certain  time,  say 
n  seconds,  adjust  the  zero,  allow  current  to  flow  n  seconds,  and 
read;  cut  off  current,  and  if  after  n  seconds  the  pointer  is  not 
exactly  on  zero,  repeat  the  operations.  The  exact  number  of 
seconds  to  be  allowed  for  each  operation  depends  on  the  particular 
instrument.  If  the  instrument  has  no  zero  adjustment,  an  accu- 
rate measurement  can  only  be  made  by  repeating  the  procedure 
followed  in  the  calibration  of  the  instrument.  If  the  calibration 
procedure  is  not  known,  the  most  accurate  method  of  reading  is 
probably  to  first  allow  the  current  to  flow  for  n  seconds  without 
reading,  cut  off,  and  after  n  seconds  read  the  zero  position  of 
pointer,  allow  current  to  flow  n  seconds  and  read,  cut  off,  and 
again  take  a  zero  reading  after  n  seconds.  These  same  methods 
of  overcoming  zero  error  apply  to  all  thermal  ammeters.  In 
instruments  having  a  nonuniforrn  scale  the  zero  reading  must  not 


150 


Circular  of  the  Bureau  of  Standards 


be  made  in  scale  divisions;  it  should  be  estimated  in  millimeters, 
and  the  correction  at  any  point  found  by  multiplying  by  the 
number  of  scale  divisions  per  millimeter  at  that  point. 


42.  CURRENT  TRANSFORMERS 


Inductance  and  Capacity  Shunting.  —  The  reason  why  it  is  difficult 
to  measure  large  currents  of  high  frequency  accurately  is  that 
ordinary  shunts  can  not  be  used.  As  has  been  pointed  out,  the 
current  divides  between  an  instrument  and  its  shunt  according  to 


I       R 

and  the  current  ratio  varies  with  frequency  unless  —  2  =  -=?.     One 

L!     K! 

way  out  of  the  difficulty  has  been  explained  in  connection  with 
thermal  ammeters,  viz,  to  make  Rt  and  R2  so  large  in  comparison 


M 


© 


FIG.  104.  —  Inductance  shunted  ammeter 

with  L!  and  L2  that  the  resistances  determine  the  currents  even 
for  radio  frequencies.  This  is  done  by  use  of  wires  of  small 
diameter  and  high  resistivity.  Another  possible  solution  is  the 
opposite  of  this,  viz,  to  make  ^  and  R2  negligible  in  comparison 
with  Lx  and  L2  at  radio  frequencies.  The  method  is,  in  short,  to 
use  inductance  shunts.  If  any  low-range  ammeter  of  small 
resistance  and  inductance  is  connected  in  series  with  a  large 
inductance  L/,  the  combination  being  shunted  by  a  small  induc- 
tance M,  and  if  resistances  are  negligible,  the  current  divides 
between  the  two  paths  according  to 


_ 
/,      M 


Radio  Instruments  and  Measurements 


It  follows  that 


L2'+M 


-*^2 


M 


Now  /ro+/2=/i,  the  total  current. 

Denoting  the  total  inductance  in  the  ammeter  circuit  by  L2, 

L3'+M=L3;  then 

A_*2  (74) 

I2    M 

In  practice  the  small  inductance  M  might  be  a  single  loop  of  wire 
or  even  a  length  of  straight  wire,  and  the  coil  L/  and  the  instru- 
ment A  would  be  kept  at  such  a  distance  from  it  that  mutual 
inductance  would  have  no  effect.  This  method  could  not  be  used 
at  low  frequencies  because  the  resistances  would  affect  the  current 
ratio. 

A  similar  shunting  scheme  is  the  use  of  capacity  shunts.  A 
large  condenser  is  inserted  in  the  circuit  The  capacity  of  this 
condenser  (C  in  Fig.  105)  is  many  times  that  of  the  main  con- 


tt 


FIG.  105. — Capacity  shunted  ammeter 

denser,  with  which  it  is  in  series.  It  is  shunted  by  a  small  condenser 
C'  in  series  with  a  low-range  ammeter.  The  arrangement  has  the 
desirable  feature  of  introducing  only  a  small  impedance  into  the 
circuit,  because  the  capacity  of  condenser  C  is  large.  The  resist- 
ances in  ammeter  and  condensers  must  be  kept  small  in  order  that 
the  shunt  ratio  may  be  invariable.  Care  is  necessary  to  prevent 
stray  capacities,  from  the  body  of  the  observer,  etc.,  from  affect- 
ing this  branch  circuit.  The  posibilities  of  this  method  have  not 
been  studied  experimentally. 

Simple  Theory  of  Current  Transformer. — Inductance  shunting 
is  not  practiced  as  shown  in  Fig.  104.  The  principle,  however, 
is  made  use  of  in  the  current  transformer.  The  instrument  cir- 
cuit is  inductively  coupled  to  the  main  circuit,  as  shown  dia- 


152 


Circular  of  the  Bureau  of  Standards 


grammatically  in  Fig.  106,  instead  of  direct  coupled  as  in  Fig.  104. 
If  the  resistance  and  inductance  of  the  ammeter  are  negligible 
and  there  is  no  appreciable  energy  loss  in  the  coils  or  the  medium, 
equation  (74)  gives  the  ratio  of  currents.  The  resistance  of  the 
usual  ammeters  is  of  the  order  of  several  ohms  and  sometimes 
can  not  be  neglected.  The  emf  induced  in  the  secondary  circuit  is 
coM/i.  This  is  opposed  by  the  inductance  L2  and  by  the  resist- 
ance R2  of  the  ammeter. 


co2M2 


T     2    /  E>  2   \ 

—  =M    T  _L  2        1 

~M2VI+co2L22>/ 


[L 


. 

M\  2C02L2: 


(75) 


(R2  being  small  in  comparison  with  wL2) . 

The  current  ratio  is  that  given  by  equation  (74) ,  with  a  small 
correction  term  added.  These  calculations  assume  sine-wave 
currents,  but  apply  to  slightly  damped  currents  as  well.  If  the 


FIG.  106. — Inductance  shunted  ammeter  in- 
ductively coupled.  Principle  of  the  current 
transformer 

logarithmic  decrement  of  the  current  is  greater  than  a  few  hun- 
dredths,  an  additional  correction  term  is  needed.  In  using  equa- 
tion (75),  the  actual  high-frequency  value  of  L2  must  be  used. 

Transformer  Without  Iron. — The  current  transformers  used  in 
the  measurement  of  radio  currents  are  of  two  types,  with  and 
without  an  iron  core.  The  iron  core  has  advantages  in  certain 
circumstances;  this  is  discussed  below.  The  simple  transformer 
without  a  magnetic  core  is,  however,  satisfactory  if  used  care- 
fully. A  form  which  has  been  found 18  successful  has  a  second  - 


18  See  reference  No.  103,  Appendix  2. 


Radio  Instruments  and  Measurements  153 

ary  winding  consisting  of  a  single  layer  of  stranded  wire  on  an 
insulating  cylinder  and  a  primary  of  one  or  more  turns  of  thicker 
stranded  wire  near  the  middle  of  the  secondary  winding.  To 
avoid  induction  from  the  leads  and  other  parts  of  the  circuit  two 
such  coils  are  used,  connected  so  as  to  give  astaticism,  and  the 
ends  of  the  primary  winding  are  brought  out  to  a  considerable 
distance.  Equation  (74)  for  the  current  ratio  has  been  found  to 
apply  to  such  a  transformer,  within  the  accuracy  of  observation, 
for  ratios  as  high  as  100  to  i.  The  ratio  may  vary  a  few  per  cent 
with  the  frequency  because  of  the  resistance  of  the  instrument 
connected  to  the  secondary.  At  frequencies  below  the  radio 
range  the  correction  becomes  very  large. 

Iron-Core  Transformer. — The  iron-core  radio  current  trans- 
former180 consists  of  a  laminated  iron  ring  with  a  close  winding 
of  one  or  a  few  layers  of  fine  wire  upon  it  and  a  small  number  of 
primary  turns  of  heavy  stranded  wire  linking  with  it,  as  shown  in 
Figs.  107  and  109,  facing  page  157.  Very  thin  silicon-iron  sheet 
is  a  satisfactory  core  material.  The  ring  may  be  very  small,  of 
the  order  of  5  cm  diameter.  Little  care  need  be  taken  to  avoid 
induction  from  other  parts  of  the  primary  circuit,  as  in  the  use 
of  the  transformers  without  iron.  The  iron  core  greatly  increases 
L2  and  insures  close  coupling  between  the  primary  and  secondary 
turns. 

The  current  ratio  is  readily  found  in  terms  of  the  ratio  of  pri- 
mary and  secondary  turns.  The  self-inductance  of  the  secondary 
winding  is  given  by 


I0»d 

where  n2  =  number  of  secondary  turns,  A  =  area  of  cross  section 
of  iron,  jua  =  apparent  permeability  of  iron  at  the  actual  frequency 
used,  and  d  =  mean  diameter  of  the  iron  ring.  The  mutual  in- 
ductance is 


io9d 
t  being  number  of  primary  turns.     It  follows  that 

L2==n2 
M~nt 

iso  See  reference  No.  103,  Appendix  2. 


1  54  Circular  of  the  Bureau  of  Standards 

Since  the  ratio  of  primary  to  secondary  current  was  found  above 

to  be  approximately  rl  it  follows  that  in  the  iron-core  trans- 

former the  current  ratio  is  approximately  the  ratio  of  turns.  The 
equation  previously  derived  for  current  ratio  does  not  apply 
exactly  to  the  iron-core  transformer  because  of  the  assumption 
of  no  energy  loss.  There  is  an  energy  loss  in  the  iron  due  to  eddy 
currents  and  hysteresis.  This  requires  an  energy  current  in  the 
primary,  which  disturbs  the  current  ratio.  Taking  account  of 
this,  the  ratio  may  be  shown  to  be 

(^,. 

(76) 


in  which  a  is  a  quantity  depending  on  the  energy  loss  in  the  iron, 
having  a  value  which  is  usually  slightly  less  than  unity.  This 
assumes  that  all  the  magnetic  flux  from  the  secondary  circuit 
links  with  the  primary  turns,  a  condition  which  is  not  fulfilled 
unless  the  secondary  is  uniformly  and  closely  wound  and  the 
inductance  of  the  instrument  connected  to  the  secondary  is 
negligible. 

Because  of  the  iron  core  the  secondary  inductance  L2  is  so 
large  that  the  correction  term  in  equation  (76)  is  ordinarily 
negligible  at  radio  frequencies.  Thus  an  advantage  of  the  iron- 
core  transformer  is  that  the  current  ratio  does  not  vary  appre- 
ciably with  frequency  and  does  not  depend  upon  the  instrument 
connected  to  it.  Careful  design  is  necessary  to  secure  this  con- 
stancy of  ratio,  and  even  then  it  holds  only  for  radio  frequencies. 
At  low  frequencies  these  current  transformers  have  large  errors. 
The  reason  is  easily  seen,  since  the  correction  term  in  equa- 
tion (76)  increases  as  w  decreases.  The  increase  of  the  correction 
is  not  proportional  to  the  decrease  of  co  because  L2  is  smaller  for 
high  frequencies  than  for  low.  The  value  of  L2  is  proportional 
to  the  apparent  permeability  of  the  iron,  which  decreases  with 
increase  of  frequency  because  the  skin  effect  reduces  the  effective 
cross  section  of  the  iron.  Thus  in  a  certain  transformer  19  the 
iron  had  a  permeability  of  1000  at  50  cycles  and  an  apparent 
permeability  of  30  at  200  ooo  cycles.  The  correction  term  in  the 
current  ratio  equation  was  14  per  cent  at  the  lower  frequency  and 
0.2  per  cent  at  the  higher. 

19  See  reference  No.  104,  Appendix  2. 


Radio  Instruments  and  Measurements  155 

The  apparent  permeability  depends  on  the  thickness  of  the  iron 
laminations.  Making  them  thinner  improves  the  accuracy  of  the 
transformer  in  two  ways — it  increases  the  apparent  permeability 
and  thus  increases  L2,  and  it  also  makes  a  larger  proportion  of  the 
magnetic  flux  from  the  secondary  link  with  the  primary  turns. 
It  is  interesting  to  note  that,  at  a  given  frequency,  the  apparent 
permeability  of  the  iron  does  not  vary  with  the  current,  because 
the  fluxes  in  the  iron  are  so  very  small  that  the  permeability  is" 
practically  constant. 

Ad-vantages  of  Transformer. — The  ring  form  of  current  trans- 
former has  also  been  used  without  the  iron  core.  Under  certain 
conditions,  as  when  an  ammeter  of  extremely  small  resistance  is 
connected  to  the  secondary,  this  may  be  a  very  good  form  of 
instrument.  Neither  this  nor  the  iron-core  transformer  has  been 
exhaustively  studied.  Both  forms,  however,  are  definitely  known 
to  have  the  following  advantages  as  devices  for  measuring  large 
currents:  (i)  They  conform  to  the  requirement  of  simplicity  of 
circuit,  for  the  primary  turns  have  very  little  inductance  and 
capacity;  (2)  they  utilize  the  magnetic  effect  of  the  current  and 
are  thus,  in  themselves,  free  from  the  inherent  limitations  of 
thermal  ammeters  such  as  thermal  lag  and  dependence  on  sur- 
rounding conditions;  (3)  the  measuring  circuit  is  electrically 
insulated  from  the  main  circuit  and  there  is  thus  no  conducting 
path  to  the  indicating  instrument  and  so  the  capacity  of  the  latter 
can  not  cause  so  great  a  loss  of  current  from  the  main  circuit. 

Volt-ammeter  Employing  Current  Transformer. — The  current 
transformer  is  used  in  a  portable  measuring  instrument  designed 
at  the  Bureau  of  Standards  for  the  use  of  the  radio  inspectors  of 
the  Bureau  of  Navigation,  Department  of  Commerce.  Two  views 
of  the  instrument  are  shown  in  Figs.  108  and  109,  facing  page  157. 
This  instrument,  which  is  called  a  volt-ammeter,  is  a  combination 
of  a  current-square  meter,  two  high-frequency  current  transform- 
ers, and  series  resistances  which  may  be  thrown  into  the  circuit 
when  it  is  desired  to  use  the  instrument  as  a  voltmeter. 

The  current-square  meter  is  of  a  standard  commercial  type, 
requiring  approximately  o.i  ampere  for  full  scale  deflection. 
Three  scales  are  provided,  running  from  o  to  100,  o  to  5,  and  o  to  25, 
respectively.  The  o  to  100  scale  is  of  use  only  in  wave  length  or 
decrement  measurement  where  relative  values  of  current  square 
are  desired.  The  other  two  scales  indicate  amperes,  and  depend 
for  their  calibration  on  the  accurate  adjustment  of  the  number  of 


156  Circular  of  the  Bureau  of  Standards 

turns  on  the  secondaries  of  the  two  current  transformers  contained 
in  each  instrument. 

Views  of  the  transformers  are  shown  in  Fig.  109.  The  cores  of 
both  transformers  are  composed  of  a  number  of  ring-shaped 
laminations  of  very  thin  silicon  steel.  These  are  bound  tightly 
together  and  served  with  a  layer  of  empire  cloth  tape.  Over  this 
is  uniformly  wound  a  single  layer  of  fine  wire  comprising  the 
secondary.  The  number  of  secondary  turns  is  determined  from 
the  relation 

n^n/f-  (77) 

where 

n2  =  number  of  secondary  turns  in  series, 
n^  =  number  of  primary  turns  in  series, 
72  =  secondary  current  (about  o.  i  ampere) , 
I1  =  primary  current,  either  5  or  25  amperes. 
In  these  transformers  the  number  of  primary  turns  (nj  adopted 

was  2,  so  that  expression  (77)  becomes 

^ 

W2~2/2 

This  relation  holds  very  closely  at  high  frequencies  if,  as  in  this 
case  of  a  toroid  winding,  the  magnetic  leakage  is  small  or  negligible. 
Small  magnetic  leakage  requires  that  the  secondary  be  wound  in 
a  single  layer  as  uniformly  as  possible,  covering  the  entire  core 
length.  The  primary  is  wound  in  the  manner  indicated  in 

Fig.  107,  facing  page ,  the  wire  being  supported  at  a  distance 

from  the  secondary  and  core.  The  terminals  of  the  primaries  of 
the  transformers  are  brought  out  to  four  large  binding  posts  at 
the  bottom  of  the  instrument.  The  secondary  terminals  are 
connected  to  opposite  sides  of  a  double-pole  double-throw  switch 
which  is  arranged  to  place  the  meter  in  series  with  either  of  the 
two  windings. 

In  operation,  when  it  is  desired  to  measure  the  antenna  current 
of  a  transmitting  set,  the  meter  is  connected  in  series  with  the 
antenna  and  ground  at  the  large  binding  posts  marked  5  or  25 
amperes,  depending  upon  the  magnitude  of  the  current  to  be  meas- 
ured. The  double-throw  switch  is  then  thrown  to  place  the 
proper  transformer  in  circuit  and  the  readings  obtained.  The 
instrument  will,  of  course,  work  equally  well  in  a  closed  circuit. 


Bureau  of  Standards  Circular  No.  74 


FIG.  96. — Hot-wire  type  current-square  meter 


FIG.  102. — Hot-strip  ammeter  -with  cylindrical  arrangement 
of  heating  elements 


FIG.  in. — Mounted  thermocouple  -with  protecting  cap  removed 


Bureau  of  Standards  Circular  No.  74 


FIG.  107. — Iron-core  current  transformer 


FIG.  108. — Volt-ammeter  employing  the  current 
transformer 


FIG.  109. — Rear  -view  of  volt-ammeter,  showing  the  current 
transformers 


Radio  Instruments  and  Measurements  157 

As  previously  mentioned,  the  accuracy  of  the  current  scales 
depends  upon  the  proper  adjustment  of  the  number  of  secondary 
turns.  It  is  also  affected  by  the  frequency,  the  error  growing 
larger  as  the  frequency  is  decreased.  These  transformers  are  so 
constructed  and  calibrated  that  working  over  a  range  of  fre- 
quencies corresponding  to  wave  lengths  between  150  and  1000 
meters  the  current  scales  are  accurate  to  better  than  2  per  cent. 

The  rubber-covered  binding  posts  at  the  top  of  the  meter,  in 
conjunction  with  the  push  button  marked  "Voltage",  are  used  for 
voltage  measurement.  Three  ranges  are  provided,  2.7, 40,  and  1 50 
volts.  These  were  adopted  so  as  to  enable  the  inspector  to  make 
proper  voltage  test  on  the  storage  batteries  of  the  auxiliary  power 
supply.  Voltage  calibration  curves  are  supplied  with  the  instru- 
ment. 

The  two  sockets  at  the  left  marked  decremeter  connect  directly 
to  the  terminals  of  the  current-square  meter.  They  are  used 
when  the  meter  is  employed  as  a  current  indicator  in  wave  length 
or  decrement  measurement. 

This  volt-ammeter  was  designed  with  the  primary  object  of 
reducing  the  weight  and  number  of  instruments  which  the  radio 
inspector  must  carry  with  him.  In  the  wavemeter  previously 
used  by  the  inspectors  the  current-square  meter  was  contained 
within  the  wavemeter  case.  In  the  latest  type  the  current- 
square  meter  has  been  omitted,  thus  providing  a  much  smaller 
and  lighter  wavemeter.  This,  together  with  the  volt-ammeter 
here  described,  provides  an  equipment  for  making  all  the  required 
measurements. 

43.  MEASUREMENT  OF  VERY  SMALL  CURRENTS 

A  number  of  methods  are  used  for  measuring  currents  of  a  few 
milliamperes  or  less.  In  addition  to  the  thermal  ammeter  and 
the  current  transformer,  used  for  larger  currents,  use  is  made 
of  the  electrostatic  and  the  magnetic  effects  of  the  current,  and 
rectification  into  unidirectional  current.  Instruments  operating 
on  these  various  principles  are  described  below.  The  measure- 
ment of  very  small  currents  is  free  from  some  of  the  difficulties  of 
measuring  larger  currents,  principally  because  conductors  of  very 
small  cross  section  may  be  used  which  do  not  change  in  resistance 
with  frequency. 

Crossed-Wire  Thermoelement. — Sensitive  thermoelements  are 
easily  made  and  are  extensively  used  to  measure  small  high-fre- 
quency currents.  They  consist  essentially  of  two  wires  of  different 


1 58  Circular  of  the  Bureau  of  Standards 

metals  in  contact,  one  or  both  of  them  being  of  very  small  diameter. 
The  heat  due  to  the  R  I2  in  the  fine  wire  raises  the  temperature 
of  the  junction,  thus  giving  rise  to  a  thermoelectromotive  force 
which  is  indicated  by  a  direct-current  galvanometer.  A  simple 
type  is  the  crossed-wire  thermoelement.  Two  fine  wires  are  used, 
crossed  in  either  of  the  ways  shown  in  Fig.  no.  (Practically  the 
same  type  was  shown  in  Fig.  98) .  The  high-frequency  current  is  led 
in  through  the  heavy  copper  wires  A  and  B,  and  the  galvanometer  is 
connected  to  a  and  b.  The  sensitivity  of  the  thermoelement 
depends  on  the  diameter,  thermoelectric  properties,  and  resistivity 
of  the  wires,  on  the  length  of  the  wires  if  very  short,  on  the  intimacy 
of  contact  of  the  two  wires,  and  on  the  air  pressure.  Permanent 
contact  of  the  two  wires  may  be  insured  by  soldering  the  junction. 
The  use  of  a  minute  particle  of  solder  does  not  appreciably  reduce 


b  B 

FIG.  no. — Crossed-wire  thermoelements 

the  sensitivity.  If  the  junction  is  not  soldered,  some  thermoele- 
ments are  occasionally  found  to  have  an  abnormally  high  sensitiv 
ity,  probably  owing  to  a  particularly  poor  contact ;  the  high  resist- 
ance of  the  contact  causes  a  production  of  heat  just  at  the  junction 
which  is  large  relatively  to  the  heat  produced  in  the  wires.  In  some 
thermoelements  this  poor  contact  remains  sufficiently  constant 
so  that  the  thermoelement  can  be  relied  upon,  and  in  some  it  does 
not. 

The  materials  ordinarily  used  for  these  thermoelements  are 
constantan  and  steel,  and  constantan  and  manganin.  Using 
wires  of  the  order  of  0.02  mm  diameter  and  4  mm  long,  a  con- 
stantan-steel  thermoelement  has  a  resistance  of  about  i  ohm 
and  gives  about  40  microvolts  for  1 5  milliamperes  high-frequency 
current.  Using  a  galvanometer  with  a  sensitivity  of  2.5  mm  per 
microvolt,  it  follows  that  a  deflection  of  100  mm  is  produced  by 
a  high-frequency  current  of  15  milliamperes.  The  electromo- 
tive force  is  very  closely  proportional  to  the  square  of  the  high- 
frequency  current.  Thus,  the  typical  thermoelement  just  men- 
tioned gives  a  deflection  of  only  i  mm  for  1.5  milliamperes. 


Radio  Instruments  and  Measurements  1 59 

These  thermoelements  have  practically  no  thermal  lag  because 
of  being  made  of  such  fine  wire;  it  therefore  pays  to  use  a  very 
quick-acting  galvanometer  with  them.  A  thermoelement  of  this 
type  made  at  the  Bureau  of  Standards  is  shown  in  Fig.  1 1 1 ,  facing 
page  156. 

Smaller  currents  can  be  measured  by  the  use  of  thermoelements 
made  of  wires  of  still  smaller  diameter.  The  resistance,  the  RP, 
and  therefore  the  temperature,  would  be  higher  for  a  given  cur- 
rent. High  resistance  is,  however,  objectionable  in  most  radio 
circuits,  so  that  the  more  sensitive  thermoelements  of  higher 
resistance  have  little  application. 

Another  way  to  increase  sensitivity  at  the  expense  of  increasing 
the  resistance  of  the  circuit  is  to  connect  the  thermoelement,  not 
directly  into  the  circuit,  but  to  the  low-voltage  side  of  a  current 
transformer  in  the  circuit.  This  has  the  advantage  that  the 
galvanometer  is  not  metallically  connected  to  the  main  circuit 
and  thus  its  capacity  is  less  likely  to  cause  leakage  of  high-fre- 
quency current  from  the  main  circuit.  The  exact  limitations  of 
this  method  are  not  known;  it  seems  likely  that  the  calibration 
will  change  with  frequency  and  with  the  resistance  and  inductance 
connected  to  the  transformer. 

The  above  figures  on  sensitivity  are  for  thermoelements  in  air 
at  ordinary  atmospheric  pressure.  No  variation  of  sensitivity 
with  the  ordinary  barometric  pressures  has  been  observed,  but 
the  sensitivity  may  be  greatly  increased  by  placing  the  thermo- 
element in  a  vacuum.  It  has  been  found  that  an  air  pressure 
of  about  o.oi  mm  of  mercury  or  lower  is  necessary  to  gain  much 
in  sensitivity,  but  that  in  such  low  vacua  the  sensitivity  of  thermo- 
elements of  polished  metal  wires  may  be  increased  as  much  as  25 
times.  The  removal  of  the  air  eliminates  the  cooling  of  the 
thermoelement  by  convection;  a  given  current  therefore  raises 
it  to  a  higher  temperature.  The  temperature  of  a  hot  body  in  a 
vacuum  is  limited  only  by  radiation  of  heat  from  its  surface; 
thus  the  temperature  of  a  polished  metal  surface,  which  is  a 
poor  radiator,  rises  higher  than  that  of  a  dull  metal  surface, 
which  is  a  good  radiator. 

Self-Heated  Thermoelement. — In  the  type  of  thermoelement 
described  in  the  preceding  section  the  high-frequency  current 
does  not  pass  through  the  wires  of  the  thermocouple  itself.  The 
thermocouple  wires  (Oa  and  Ob,  Fig.  no)  touch  the  wires 
(OA  and  OB),  which  carry  the  high-frequency  current,  at  one 

35601°— 18 11 


1 60  Circular  of  the  Bureau  of  Standards 

point  only.  There  is  no  reason  why  the  thermocouple  wires 
themselves  may  not  carry  the  high-frequency  current  and  heat 
the  junction.  A  simple  thermocouple  so  used  may  be  called  a 
self-heated  thermoelement.  By  using  this  type,  Austin20  has 
found  it  possible  to  utilize  tellurium  as  one  of  the  metals  of  a 
thermoelement  and  obtain  very  high  sensitivity.  A  couple 
consisting  of  tellurium  and  platinum  gives  about  25  times  as 
great  an  emf  as  constantan  and  platinum  at  the  same  temperature. 
These  thermoelements  are  made  by  the  following  process. 

Two  copper  wires  are  placed  side  by  side  about  3  mm  apart 
and  embedded  in  insulating  material  with  their  ends  protruding 
(Fig.  112).  To  the  end  of  one  of  these  is  soldered  about  5  mm 
of  0.02  mm  platinum  or  constantan  wire.  To  the  other  is  soldered 
a  short  bit"  of  0.8  mm  platinum  wire,  to  the  end  of  which  a  bead 
of  tellurium  had  previously  been  attached  when  the  platinum 
was  white  hot.  (White-hot  platinum  wire  when  inserted  in 
tellurium  forms  practically  a  resistance-free 
contact.)  The  end  of  the  fine  wire  is  next 
allowed  to  rest  against  the  tellurium  and  the 
two  are  welded  together  electrically  by  means  of 
a  small  induction  coil  with  a  high  resistance 
in  series  with  the  secondary.  The  contact  will 

FIG.  ^.-Platinum-  be  less  fra§ile  if  the  welding  is  done  in  an  oxy- 
tellurium  thermocou-  gen-free  atmosphere.  The  resistance  of  a  ther- 
pie  of  the  self-heated  moelement  prepared  in  this  way  may  be  any- 
where from  5  to  50  ohms,  according  to  the  condi- 
tions of  welding  and  the  resistance  of  the  fine  wire,  the  lower 
resistance  being  somewhat  more  difficult  to  obtain.  The  thermo- 
element is  next  inclosed  in  a  test  tube,  and,  if  likely  to  be  handled 
roughly,  the  whole  may  be  inclosed  in  a  larger  test  tube  with 
cotton  or  felt  between  the  two.  These  thermoelements  remain 
constant  over  considerable  periods,  but  some  have  been  found  to 
lose  their  sensitiveness  after  a  year  or  two.  A  3 2 -ohm  thermoele- 
ment was  found  to  give  a  deflection  on  a  very  sensitive  galva- 
nometer of  i  mm  for  120  microamperes.  Such  thermoelements 
are  very  satisfactory  when  used  in  connection  with  portable 
microammeters  of  the  pointer  type. 

Thermoelements  of  the  self-heated  type,  can,  of  course,  not  be 
used  on  direct  current,  since  the  galvanometer  forms  a  shunt  on 
the  heating  wire.  At  high  frequencies  the  current  is  kept  out  of 
the  galvanometer  because  its  impedance  is  so  much  greater  than 

20  See  reference  No.  207,  Appendix  2. 


Radio  Instruments  and  Measurements 


161 


that  of  the  short  lengths  of  thermocouple  wires  which  it  shunts. 
It  should  be  noted  that  the  heating  current  passes  through  both 
junctions  of  the  thermocouple  but  that  the  temperature  of  only 
one  rises  appreciably  because  a  fine  wire  of  high  resistance  is 
used  at  one  junction  only.  Large  cross  section  at  the  other 
junction  prevents  a  large  heat  production  and  temperature  rise. 
Thermogahanometer. — Currents  of  several  hundred  microam- 
peres may  be  conveniently  measured  by  the  Duddell  thermo- 
galvanometer.  This  is  a  compact  combination  of  hot  wire, 
thermocouple,  and  galvanometer.  The  galvanometer  coil  is  a 
single  slender  turn  of  wire,  with  the  bismuth-antimony  thermo- 
couple attached  to  its  lower  end.  The  junction  between  these 


Bi  VSb 


Hot  wire 

FlG.  113. — Duddell  thermogahanometer 

two  metals  is  directly  over,  but  not  in  contact  with,  the  heater 
which  is  either  a  hot  wire  or  thin  gold  leaf  or  film  of  platinum  on 
glass.  Currents  as  low  as  10  microamperes  may  be  measured 
when  a  heater  of  several  thousand  ohms  resistance  is  used,  but 
this  great  sensitivity  is  not  available  for  radio  measurements, 
which  generally  require  heaters  of  less  than  50  ohms  resistance. 
The  thermogalvanometer  differs  from  the  thermoelements  de- 
scribed above  in  that  the  thermocouple  is  not  in  metallic  contact 
with  the  heater,  and  thus  the  capacity  of  the  galvanometer  is 
less  likely  to  cause  leakage  of  high-frequency  current  from  the 
main  circuit. 

Bolometer. — An  instrument  in  which  heat  is  measured  by  the 
change  of  resistance  which  it  produces  in  a  conductor  is  called  a 
bolometer.  Precise  measurements  of  radiant  heat,  for  instance, 


l62 


Circular  of  the  Bureau  of  Standards 


are  made  with  a  bolometer  which  consists  of  a  blackened 
metal  strip  together  with  a  Wheatstone  bridge  for  measuring  its 
resistance.  The  bolometer  used  in  radio  measurements  consists 
essentially  of  a  fine  wire,  through  which  the  high-frequency  cur- 
rent is  passed,  connected  to  a  Wheatstone  bridge.  The  resistance 
of  the  wire  increases  as  it  is  heated  by  the  current,  and  this  change 


FlG.  114. — Various  methods  of  using  the  bolometer  for  measuring  small  radio  currents 

of  resistance  either  causes  a  deflection  of  the  bridge  galvanometer, 
or  the  bridge  is  balanced  by  a  change  in  one  of  the  bridge  arms 
so  as  to  keep  the  galvanometer  deflection  unchanged.  In  the 
latter  method  the  resistance  in  the  changed  bridge  arm  is  a  measure 
of  the  current. 

The  direct  current  used  in  the  Wheatstone  bridge  produces 
more  or  less  heat  in  the  bolometer  wire,  and  in  order  to  avoid 


Radio  Instruments  and  Measurements  1 63 

error  it  is  necessary  to:  (i)  Keep  it  very  small,  or  (2)  keep  it 
constant  for  all  measurements,  or  (3)  have  an  auxiliary  wire 
similar  to  the  bolometer  wire  in  an  adjacent  arm,  which  will  keep 
the  bridge  balanced  as  far  as  heating  by  the  direct  current  in  the 
bridge  is  concerned.  The  various  forms  shown  in  Fig.  114  have 
been  used.  Figures  (6)  and  (d)  are  similar  to  (a)  and  (c),  re- 
spectively, except  for  the  auxiliary  wires  in  a  bridge  arm  adjacent 
to  the  one  containing  the  bolometer  wire.  It  is  necessary  to 
keep  the  high-frequency  current  out  of  the  parts  of  the  bridge 
other  than  the  bolometer  wire  as  the  latter  would  not  then  carry 
the  whole  current  to  be  measured.  This  is  done  in  either  of  the 
two  ways  shown.  Choke  coils  may  be  used  on  either  side  of  the 
hot  wire,  as  in  (a)  and  (6);  or,  the  bolometer  wire  may  have 
the  rhombus  form  as  in  (c)  and  (d).  The  heating  current  divides 
between  the  two  halves  of  the  rhombus,  and  the  bridge  connections 
are  made  at  two  points  of  equal  potential  so  that  there  is  no 
tendency  for  the  heating  current  to  flow  into  the  bridge. 

For  the  bolometer  wire,  use  has  been  made  of  iron,  gold,  plati- 
num, tungsten,  and  carbon.  The  smaller  the  wire  the  smaller  the 
currents  measurable.  A  gold  wire  0.002  mm  diameter  has  been 
found  to  give  10  scale  divisions  on  a  pointer- type  galvanometer  for 
500  microamperes  and  a  0.0005  mra  platinum  wire  10  scale  divi- 
sions on  the  same  galvanometer  for  34  microamperes.  These  fig- 
ures 21  are  for  the  bolometer  wire  in  air.  The  sensitivity  may  be 
increased  by  placing  in  a  vacuum.  A  current  of  5  microamperes 
has  been  measured  by  the  use  of  a  carbon  filament  in  a  vacuum. 

The  bolometer  has  also  been  used  as  a  means  of  measuring  large 
currents  up  to  10  amperes.  (Seep.  172  below.)  Wire  of  com- 
paratively large  diameter  is  used  but  not  so  large  as  to  change  in 
resistance  with  frequency.  It  is  immersed  in  oil  to  keep  down  the 
the  temperature  rise. 

Electrometer. — It  is  possible  to  measure  fairly  small  currents 
by  the  aid  of  an  electrometer  shunted  across  a  condenser.  The 
deflections  of  an  electrometer  are  proportional  to  the  square  of 
the  effective  voltage  when  the  vane  is  connected  to  one  plate. 
The  current  through  the  electrometer  is  proportional  to  its  capa- 
city, to  the  frequency,  and  to  the  applied  voltage.  The  form 
shown  in  Fig.  116  has  been  used.  Connection  is  made 
to  the  fixed  plates  PP.  The  light  metal  vane  V  is  suspended 
by  a  delicate  fiber  between  them.  The  suspending  fiber  also 

21  Data  from  B.  Gati  (Electrician,  58,  p.  983,  1907,  and  78,  p.  354,  1916),  who  uses  an  arrangement  for 
measuring  resistance  somewhat  different  from  a  Wheatstone  bridge,  and  who  calls  his  device  a ' '  barretter." 


164 


Circular  of  the  Bureau  of  Standards 


carries  a  damping-vane  and  a  mirror  for  reading  from  a  distance. 
The  voltage  impressed  on  the  plates  PP  charges  them  with  electric- 
ity of  opposite  signs.  One  of  them  being  connected  to  the  vane 
V,  they  thus  exert  a  torque  on  it. 


FlG.  115 . — Method  of  using  electrometer  (vane 
should  be  connected  to  one  terminal) 

These  instruments  have  the  advantage  of  introducing  no  appre- 
ciable resistance  into  the  circuit.  They  have  a  small  capacity  and 
therefore  should  not  be  used  in  parallel  with  any  except  a  large 
condenser.  They  require  very  careful  manipulation,  as  they  are 

delicate  and  are  very  sensi- 

^/"~V-\     *lve    *°    s^ray   electrostatic 

IV    ~^~~\A     charSes-    The   method   has 

^— '  not  been  used  in  the  Bureau 

FIG.  116.— Schematic  ar-  of  Standards  laboratory. 

rangement  of   plates      Electrodynamometer. — The 

and  -vane  of  the  electro-   instruments  of  this  type  used 
meter  J  \ 

at  low  frequency  consist  of  a 

fixed  and  moving  coil  connected  either  in  series 
or  in  parallel.  Capacity  between  the  two  coils 
renders  them  unsuitable  for  measuring  high- 
frequency  currents.  A  somewhat  different  type 
has,  however,  been  successfully  used.  The  in- 
strument consists  of  a  small  coil  of  fine  wire 
which  carries  the  current  to  be  measured,  and  pIG  II7.—Typeofel£c- 
a  flat  ring  or  disk  of  silver  or  copper  suspended 
by  a  delicate  fiber  concentric  with  the  coil  and 
with  its  plane  at  45°  to  the  plane  of  the  coil. 
Current  in  the  coil  induces  an  opposing  current  in  the  ring,  which 
is  then  repelled.  The  torque  acting  on  the  ring  is  proportional  to 

LNtfP 


trodynamometer  suit- 
able for  radio  current 
measurement 


Radio  Instruments  and  Measurements 


165 


where  R  and  L  are  the  resistance  and  inductance  of  the  metal 
ring  and  N  the  number  of  turns  in  the  coil.  The  deflection  will 
therefore  depend  on  the  frequency,  which  is  a  serious  limitation 
on  the  usefulness  of  the  instrument.  For  frequencies  so  high,  how- 
ever, that  R  is  small  compared  with  coL,  the  deflections  are  inde- 
pendent of  frequency.  Care  should  be  taken  to  keep  the  instru- 
ment away  from  other  parts  of  the  circuit,  so  that  no  large  mag- 
netic fields  may  act  upon  it. 

Crystal  Detector, — Currents  too  small  to  be  measured  by  any  of 
the  preceding  instruments,  as  for  example  the  received  currents 
in  antennas,  can  be  measured  by  the  aid  of  a  crystal  detector. 


1 

c      1 

1  1 

1 

I 

U  . 

FIG.  118. — Crystal  detector  circuits  used 
in  measuring  small  currents 

The  exact  action  of  these  detectors  is  a  complicated  matter,  but 
for  practical  purposes  it  is  sufficient  to  regard  them  as  unilateral 
conductors ;  that  is,  they  have  a  greater  resistance  to  current  flow- 
ing through  them  in  one  direction  than  to  current  flowing  in  the 
opposite  direction.  Thus,  when  an  alternating  emf  is  impressed 
on  a  crystal  detector,  more  current  flows  in  one  direction  than  in 
the  other,  and  a  direct-current  instrument  in  the  circuit  will  be 
operated.  The  resistance  of  the  ordinary  crystal  detectors  in  the 
low-resistance  direction  is  of  the  order  of  1000  to  10  ooo  ohms, 
and  the  resistance  in  the  opposite  direction  about  10  times  as 
great. 


1 66  Circular  of  the  Bureau  of  Standards 

In  combination  with  a  sensitive  galvanometer,  a  crystal  detector 
may  be  used  to  measure  currents  of  a  few  microamperes.  The 
galvanometer  and  crystal  may  be  connected  to  the  circuit  LC,  the 
current  in  which  is  to  be  measured,  in  any  of  the  ways  shown  in 
Fig.  1 1 8.  Using  the  first  mode  of  connection,  Austin  22,  has  ob- 
tained a  deflection  of  100  mm  for  91  microamperes.  A  perikon 
detector  (chalcopyrite-zincite)  was  used,  with  a  galvanometer  of 
2000  ohms  resistance  giving  i  mm  for  1.3  x  io~9  ampere  direct  cur- 
rent. The  deflections  have  been  found  to  be  proportional  to  the 
square  of  the  high-frequency  current  (for  this  and  some  other  crys- 
tals) .  The  sensitiveness  of  the  crystal  detector  is  hundreds  of  times 
that  of  the  thermoelement,  but  it  is  not  constant.  It  is  always 
calibrated  just  before  or  after  use  (or  both)  by  comparison  with  a 
thermoelement  in  the  LC  circuit,  using  current  from  a  buzzer  or 
other  source.  (See  p.  1 74  below.) 

A  telephone  may  be  used  in  place  of  the  galvanometer  in  any  of 
the  arrangements  shown  in  Fig.  118,  when  periodically  interrupted 
current  is  to  be  measured.  Telephone  measurements  can  not 
be  made  of  uninterrupted  undamped  currents.  Quantitative 
measurements  may  be  made  with  the  telephone  in  two  ways.  In 
both,  the  current  through  the  telephone  is  reduced  until  the  sound 
is  just  barely  audible.  (The  limit  of  audibility  is  sometimes  taken 
to  be  that  at  which  dots  and  dashes  can  just  barely  be  dis- 
guished.)  In  the  first  method  a  resistance  is  placed  in  parallel 
with  the  telephone  and  reduced  until  the  limit  of  audibility  is 
reached;  this  is  the  "shunted  telephone"  method.  The  second 
method  employs  variable  coupling  between  the  detector  circuit 
and  the  main  circuit. 

A  measure  of  current  in  the  shunted  telephone  method  is 
obtained  as  follows:  If  t  is  the  impedance  of  the  telephone  for  the 
frequency  and  wave  form  of  the  current  impulses  through  it,  s  the 
impedance  of  the  shunt,  7t  the  least  current  in  the  telephone  which 
gives  an  audible  sound,  and  /  the  total  current  flowing  in  the  com- 
bination of  telephone  and  shunt, 

I  _s  +  t 

/t       s 

s  +  t 
This  ratio,  -  — ,  is  called  the  audibility.     It  is  approximately  pro- 

o 

portional  to  the  square  of  the  high-frequency  current.  It  can  be 
expressed  in  units  of  current  if  calibrated  at  one  or  more  values 
of  current  by  some  deflection  device  in  the  high-frequency  circuit. 

22  See  reference  No.  206,  Appendix  2. 


Radio  Instruments  and  Measurements  1 67 

The  observed  settings  depend  on  the  frequency,  on  the  wave  form 
of  the  pulses  passing 'through  the  telephone,  on  the  constants  of 
the  circuit,  and  on  the  frequency  of  interruption  of  the  current 
used,  and  involve  the  assumption  that  the  crystal  and  the  sensi- 
tiveness of  the  operator's  ear  remain  constant.  It  is  desirable  to 
minimize  the  variations  of  resistance  which  have  to  be  made  in 
the  detector  circuit  by  using  a  fixed  resistance  J?!  in  series  with  the 
detector  and  shunt  the  telephone  across  a  variable  portion  R2  of 
this.  The  apparatus  should  be  calibrated  under  the  exact  condi- 
tions of  use,  both  before  and  after  each  set  of  measurements. 

An  accuracy  of  10  per  cent  is  difficult  to  obtain;  the  method  is 
nevertheless  very  useful  in  measurements  of  radio  currents  in  re- 
ceiving sets.  For  a  frequency  of  interruption  of  the  radio  current 
of  looo  per  second,  currents  of  the  same  order  of  magnitude  can 

Ri 


— M 

FlG.   119. — Circuit  for  measuring  audi- 
bility ratios 

be  measured  with  a  crystal  detector  by  the  use  of  a  telephone  as 
by  the  use  of  a  galvanometer.  About  10  microamperes  is  the 
least  current  which  can  be  detected  by  the  ordinary  crystal  and 
telephone. 

In  the  variable  coupling  method  the  telephone  is  not  shunted, 
but  the  coupling  between  the  detector  circuit  and  the  main  circuit 
is  varied  until  the  sound  in  the  telephone  is  just  barely  audible. 
The  greater  the  high-frequency  current  to  be  measured  the  looser 
is  the  coupling.  The  arrangement  is  calibrated  by  making  at 
least  one  simultaneous  observation  of  the  coupling  for  barely 
audible  sound  and  current  as  measured  by  some  other  device,  such 
as  a  previously  calibrated  crystal  and  galvanometer  connected  to 
the  main  circuit,  together  with  the  plotting  of  a  curve  between 
coupling  and  current  in  the  main  circuit.  The  coupling  may  be 
measured  in  any  arbitrary  way,  as  by  the  distance  apart  of  the 
coupling  coils. 

Audion. — The  audion  (described  below  in  section  56)  may  be 
used  for  measurements  of  current,  just  as  the  crystal  detector  is, 


1 68 


Circular  of  the  Bureau  of  Standards 


in  conjunction  with  either  a  telephone  or  a  galvanometer.  With 
the  ordinary  audion  connections,  as  in  Fig.  120,  the  sensitivity  is 
about  the  same  as  that  of  the  best  crystal  detectors.  The  actions 
of  the  audion  and  other  electron  tubes  as  detectors,  amplifiers,  etc., 


pAMMMA 


FIG.  120. — Use  of  the  audion  for  measuring  small  currents  in 
terms  of  audibility  ratios 

are  discussed  on  pages  204  to  2 10.  The  connections  shown  here  are 
for  the  shunted  telephone  method.  The  variable  coupling  method 
can  also  be  used.  In  the  figure,  L  is  the  coil  used  for  coupling  to 
the  circuit  in  which  the  current  is  to  be  measured,  Cl  is  a  small 


FIG.  121. — Oscillating  ullraudion  circuit  used  with  crystal  detector 
and  galvanometer  for  measuring  small  currents 

fixed  condenser,  and  T  the  telephone  shunted  by  the  variable  R. 
The  audibility  is  approximately  proportional  to  the  square  of  the 
high-frequency  current,  as  in  the  case  of  the  crystal  detector. 

A  galvanometer  can  be  used  with  the  audion,  but  it  must  not 
be  put  directly  in  place  of  the  shunted  telephone,  because  a  con- 


Radio  Instruments  and  Measurements 


169 


tinuous  current  would  flow  through  it  from  the  B  battery.  One 
arrangement  is  to  place  the  primary  of  a  transformer  in  series  with 
the  telephone  and  connect  to  its  secondary  a  crystal  detector  in 
series  with  a  sensitive  galvanometer.  The  changes  in  current 
which  affect  the  telephone  give  rise  to  alternating  currents  in  the 
secondary  which  are  rectified  by  the  crystal  detector  and  thus 
cause  a  deflection  of  the  direct-current  galvanometer.  This 
arrangement  is  particularly  advantageous  when  the  oscillating 
ultraudion  connections  are  used.  (For  description  of  the 
ultraudion  see  section  58  below.)  The  connections  are  given 
schematically  in  Fig.  121.  This  is  suitable  for  the  measurement 
of  undamped  currents.  The  note  in  the  telephone  T  is  produced 


FIG.  122. — Oscillating  audion  circuits  for  quantitative  measurements  on  undamped  waves 

from  distant  radio  stations 

by  the  beats  between  the  impressed  and  the  local  currents.  The 
condenser  C4  must  be  adjusted  for  maximum  deflection  of  the 
galvanometer.  Austin23  has  found  that  the  deflections  are  pro- 
portional to  the  square  of  the  high-frequency  current,  which  means 
that  the  current  in  the  telephone  is  proportional  to  the  first  power 
of  the  high-frequency  current.  (This  law  holds  only  for  the 
oscillating  condition.  When  the  audion  is  not  oscillating,  the 
deflections  are  approximately  proportional  to  the  fourth  power 
of  the  high-frequency  current.)  This  constitutes  a  method  of 
remarkable  sensitiveness  for  measuring  small  high-frequency 
currents.  Austin  found  that  for  signals  of  minimum  audibility 
on  the  simple  audion,  the  oscillating  ultraudion  gave  audibilities 

23  See  first  article  in  reference  No.  108,  Appendix  2. 


1 70  Circular  of  the  Bureau  of  Standards 

from  300  to  1000  times  as  great;  that  is,  it  would  measure  currents 
hundreds  of  times  as  small. 

For  convenience  in  measuring  received  radio  currents  from 
distant  stations  the  shunted  telephone  is  used  in  connection  with 
the  oscillating  ultraudion.  The  arrangement  shown  in  Fig.  122 
has  been  used  by  Austin.24  The  shunt  s  is  used  on  the  telephone 
T2.  The  audibility  is  approximately  proportional  to  the  current 
in  the  antenna.  The  sensitivity  is  always  measured  at  the  time 
of  use  by  comparison  with  a  silicon  detector  and  galvanometer, 
which  combination  is  in  turn  calibrated  by  comparison  with  a 
thermoelement.  This  arrangement  has  been  used  to  make  quan- 
titative measurements  on  undamped  waves  from  radio  stations 
4000  miles  away,  the  least  high-frequency  current  detectable  in 
the  receiving  antenna  being  4  x  io~9  ampere. 

44.  STANDARDIZATION  OF  AMMETERS 

The  instruments  for  high-frequency  current  measurement  may 
be  grouped  as  follows,  from  the  standpoint  of  standardization: 
(i)  Those  whose  deflections  are  the  same  at  all  frequencies,  such 
as  suitably  designed  instruments  of  the  hot-wire  type;  (2)  those 
whose  deflections  are  accurately  calculable  at  all  frequencies,  such 
as  electrometer  ammeters;  (3)  those  which  are  constant  at  all 
radio  frequencies  but  not  at  lower  frequencies,  such  as  properly 
designed  current  transformers;  and  (4)  those  which  have  to  be 
calibrated  at  the  particular  frequency  used,  such  as  the  electro- 
dynamometer,  crystal  detector,  and  audion.  Only  the  first  and 
second  of  these  groups  are  suitable  to  serve  as  standards  for  the 
calibration  of  high-frequency  ammeters.  The  second  group, 
electrometers,  is  not  actually  used  for  this  purpose,  so  the 
ultimate  standards  used  in  practice  are  instruments  of  the  hot- 
wire type. 

Small  and  Moderate  Currents. — The  instruments  described  in 
subsection  a  above  under  this  head  are  all  simple  hot-wire  types. 
If  properly  constructed,  if  the  hot  wire  is  fine  enough,  and  the 
design  is  otherwise  correct,  they  are  themselves  standards  and 
need  no  calibration  at  high  frequency.  Such  instruments  are 
calibrated  at  low  frequency  (50  to  3000  cycles  per  second).  In 
no  case  should  they  be  used  without  calibration.  While  thermo- 
couples and  some  detectors  give  deflections  approximately  propor- 
tional to  the  square  of  the  current,  they  do  not  follow  this  law 

24  See  article  in  "{Electrician,"  reference  No.  108,  Appendix  2. 


Radio  Instruments  and  Measurements  171 

exactly.  In  some  cases  the  calibration  can  be  made  with  direct 
current.  In  other  cases  this  is  not  desirable;  in  the  thermoelec- 
tric ammeter  of  the  crossed-wire  type,  for  instance,  the  Peltier 
effect  at  the  junction  causes  a  current  through  the  galvanometer, 
and  in  addition  some  of  the  direct  current  passes  through  the 
galvanometer  inasmuch  as  the  junction  has  some  resistance  and 
thus  the  galvanometer  and  the  junction  constitute  two  parallel 
paths  for  the  current.  Both  of  these  effects  may  be  eliminated 
by  reversing  the  direct  current  and  taking  the  mean  deflection 
of  the  galvanometer  (not  the  mean  reading,  in  case  the  scale  is 
calibrated  in  terms  of  the  heating  current) .  Reversing  the  direct 
current  is  equivalent  to  using  alternating  current  for  calibration. 

The   use   of   direct   current   involves   another   possible   error, 
leakage  to  the  galvanometer,  which  may  or  may  not  be  reversed 
when  the  current  is  reversed.     It  is  on 
the  whole  good  practice  to  use  alter- 
nating current  rather  than  direct  for 
standardizing  high-frequency  ammeters, 
using   as  comparison   instruments  any 
reliable  low-frequency  ammeters. 

If  there  is  any  doubt  as  to  a  high- 
frequency  ammeter's  independence   of 

frequency,  it  should  be  compared  with  o     o 

a  reliable  standard  at  radio  frequencies  |  LOW  FREQ. 

by  the  methods  given  below. 

Large  Currents.— The  design  of  most  ^  ^--Method  of  testing  am- 

meters  for  the  effect  of  change  of 

ammeters  for  large  currents  of  high  fre-     frequency 
quency  is  such  that  it  is  not  safe  to 

assume  them  independent  of  frequency.  They  should  be  standard- 
ized by  comparison  with  instruments  known  to  be  reliable  at 
several  radio  frequencies.  The  comparison  is  made  as  indicated 
in  Fig.  123.  The  instrument  to  be  tested,  X,  is  in  series  with  a 
standard  instrument,  N,  and  their  deflections  are  simultaneously 
observed  when  supplied  alternately  with  high-frequency  and  low- 
frequency  current.  The  high-frequency  circuit  LC  is  coupled  to 
a  source  of  current  such  as  a  spark  or  arc  set  or  a  pliotron,  and 
the  low-frequency  current  is  obtained  from  an  alternator  through 
a  step-down  transformer  and  a  rheostat. 

The  two  ammeters  could,  of  course,  be  compared  using  the 
high-frequency  current  only,  but  this  would  give  no  information 
as  to  the  change  of  reading  from  low  to  high  frequency.  Also, 
the  variation  of  the  readings  at  different  radio  frequencies  would 


172  Circular  of  the  Bureau  of  Standards 

not  be  as  accurately  obtained.  The  particular  advantage  of 
using  an  auxiliary  low-frequency  comparison  current  is  that  it 
enables  one  to  determine  accurately  the  difference  between  the 
high  and  low-frequency  readings  independently  of  zero  shift, 
temperature  variation,  and  other  accidental  errors.  The  experi- 
mental procedure  is  to  pass  high-frequency  current  through  the 
two  ammeters  for  a  certain  length  of  time,  say,  one  minute, 
recording  the  deflections,  then  quickly  throw  the  switch  (5  in 
Fig.  123)  and  allow  an  approximately  equal  low-frequency  cur- 
rent to  flow  the  same  length  of  time,  recording  the  deflections; 
then  high  frequency  again,  then  low  frequency,  and  finally  high 
frequency  again.  Thus,  three  high-frequency  observations  are 
obtained  with  two  low-frequency  observations  sandwichedbetween 
them,  and  errors  from  thermal  or  other  drifts  are  eliminated. 
The  lack  of  constancy  of  a  spark  or  arc  source  limits  the  precision 


FlG.  124. — Standard  ammeter  using  the  bolometer  principle 

of  an  observation  to  a  few  tenths  per  cent.  This  can  be  excelled 
with  a  pliotron  source. 

Several  variations  of  the  hot-wire  principle  are  available  as 
standard  ammeters.  The  simplest  is  a  single  wire  as  used  in 
various  instruments  previously  described,  and  capable  of  measuring 
up  to  about  2  amperes.  An  ammeter  of  about  10  amperes  range 
can  be  calibrated  by  comparison  with  such  a  standard  at  the  lower 
end  of  its  range,  2  amperes  or  less.  Such  a  calibration  is,  of  course, 
not  so  satisfactory  as  one  covering  the  whole  range. 

As  a  standard  ammeter  for  measurements  up  to  10  amperes 
an  application  of  the  bolometer  principle  may  be  used.  The 
instrument  consists  essentially  of  a  fine  copper  wire  soldered 
to  four  upright  posts,  two  of  which  carry  the  high-frequency 
current  and  the  other  two  connect  to  a  Wheatstone  bridge  as 
in  Fig.  124.  The  wire  rhombus  is  placed  in  oil.  The  high- 
frequency  current  has  two  paths  in  the  instrument  and  hence 


Radio  Instruments  and  Measurements  1 73 

great  care  is  necessary  to  insure  that  the  resistances  and  induct- 
ances of  the  two  paths  are  equal.  With  the  most  careful  con- 
struction the  current  distribution  between  the  two  paths  doubtless 
varies  somewhat  with  frequency,  but  it  is  to  be  noted  that  the 
resistance  depends  upon  the  heat  production  in  the  whole  instru- 
ment and  not  on  that  in  one  branch  only.  Small  changes  of 
current  distribution  do  not  appreciably  affect  the  resistance  of  the 
instrument,  and  it  is  consequently  a  perfectly  reliable  high- 
frequency  standard  if  carefully  constructed. 

A  curve  i's  plotted  between  the  measured  current  and  the 
resistance  in  the  rheostat  arm  of  the  Wheatstone  bridge.  In 
the  arrangement  shown  diagrammatically  in  Fig.  124,  K  is  a 
tapping  key  in  the  battery  circuit.  A  closed  galvanometer 
circuit  is  used,  thus  eliminating  errors  of  false  zero.  G  is  a  sen- 
sitive moving-coil  galvanometer.  The  current  through  the 
fine  wire  from  the  bridge  battery  should  be  o.oi  of  the  heating 
current  or  less.  It  is  not  convenient  to  calibrate  this  standard 
on  direct  current,  although  it  is  theoretically  possible  to  do  so. 
For  a  heating  current  entering  at  L  and  M,  X  and  Y  need  to  be  so 
adjusted  as  to  be  equipotential  points;  then  no  portion  of  the 
heating  current  flows  in  the  bridge  used  to  measure  the  resistance. 
However,  it  is  difficult  to  make  this  adjustment  exactly,  and  it  is 
moreover  unnecessary,  as  a  calibration  by  low-frequency  alter- 
nating current  is  just  as  good  as  a  direct-current  calibration. 
Consequently,  the  points  X  and  Y  are  simply  made  approximately 
equipotential  points,  but  not  adjustable. 

For  currents  greater  than  10  amperes,  standard  ammeters 
of  the  cylindrical  type,  described  on  page  146  above,  are  used.  As 
there  stated,  the  design  of  these  instruments  by  no  means  insures 
their  accuracy.  If  properly  constructed  of  high-resistance  metal, 
instruments  of  this  type  can  be  used  as  standards  for  moderate 
ranges  of  current  and  frequency.  Standard  instruments  for  the 
largest  currents  used  in  radio  work  are  now  under  investigation 
at  the  Bureau  of  Standards. 

Very  Small  Currents. — Thermoelements  are  usually  used  as 
standards  in  calibrating  the  instruments  used  to  measure  very 
small  currents.  As  already  explained,  it  is  in  general  better  to 
standardize  them  with  low-frequency  alternating  current  than 
with  direct  current.  In  particular,  thermoelements  of  the  self- 
heated  type  can  not  be  used  on  direct  current.  It  is  possible  to 
use  a  bolometer  as  a  standard,  and  it  likewise  should  be  calibrated 
with  low-frequency'  alternating  current,  since  if  direct  current 


174 


Circular  of  the  Bureau  of  Standards 


were  used  some  of  it  would  be  very  likely  to  get  into  the  bridge 
galvanometer.  The  thermogalvanometer  and  the  electrometer, 
on  the  other  hand,  may  theoretically  be  calibrated  with  direct 
current  but  the  experimental  errors  are  likely  to  be  larger  than 
with  low-frequency  alternating  current. 

In  the  use  of  the  crystal  detector  and  the  audion  to  measure 
current,  with  either  a  galvanometer  or  telephone,  it  is  necessary 
to  calibrate  the  arrangement  at  the  time  of  use.  These  very 
sensitive  devices  are  variable  with  time  and  with  the  conditions 
of  the  circuits.  The  calibration  of  a  crystal  detector  is  made  by 
placing  a  thermoelement  directly  in  the  circuit  to  which  the 
crystal  is  coupled  or  attached,  as  in  Fig.  125,  and  currents  used 


FIG.  125. — Method  of  calibrating  a  detector  in  terms  of  a  thermocouple 

such  as  to  give  small  deflections  on  the  thermoelement  galvano- 
meter. These  observations  fix  the  value  of  current  for  one  or 
more  points  corresponding  to  large  deflections  of  the  galvano- 
meter attached  to  the  crystal.  The  law  of  variation  of  deflection 
of  the  latter  galvanometer  with  respect  to  current  in  the  main 
circuit  is  determined  by  a  separate  experiment  in  which  the 
detector  is  coupled  more  loosely  to  the  main  circuit  so  that  the 
deflections  of  the  two  galvanometers  are  more  nearly  equal. 
In  Fig.  125,  L0  is  an  inductance  used  to  couple  the  main  circuit 
to  the  source  of  current,  Th  is  the  thermoelement,  and  Gt  its 
galvanometer.  L2  is  used  to  couple  the  detector  circuit  to 
L!,  C2  is  a  fixed  condenser,  and  G2  is  the  detector  galvanometer. 
Ordinarily  the  deflections  of  the  two  galvanometers  are  approxi- 
mately proportional,  and  the  ratio  between  the  small  thermoele- 
ment deflection  and  the  large  crystal  deflection  obtained  in  the 


Radio  Instruments  and  Measurements  175 

calibration  is  used  as  a  multiplier  to  obtain  the  equivalent  thermo- 
element deflection  from  any  smaller  crystal  deflection  subse- 
quently observed.  The  currents  so  measured  thus  depend, 
in  a  certain  sense,  on  an  extrapolation.  Great  accuracy  is  not 
expected  nor  required.  When  an  oscillating  ultraudion  is  used 
it  is  calibrated  by  comparison  with  a  detector  in  the  same  way  as 
the  detector  is  calibrated  in  terms  of  the  thermoelement.  The 
exceedingly  small  currents  measured  with  the  oscillating  ultra- 
udion thus  depend  upon  two  extrapolations. 

RESISTANCE  MEASUREMENT 
45.  HIGH-FREQUENCY  RESISTANCE  STANDARDS 

Standards  of  resistance  are  required  for  some  of  the  methods  of 
resistance  measurement  described  below.  The  resistance  of  the 
standard  must  be  accurately  known  at  all  frequencies,  and  it  is 
very  desirable  to  have  it  remain  constant  over  all  the  frequencies 
at  which  it  may  be  used.  This  requirement  practically  limits  the 
form  of  such  a  standard  to  a  very  fine  wire.  Very  thin  tubes 
would  be  satisfactory  from  the  theoretical  standpoint,  but  it  is 
extremely  difficult  to  obtain  very  thin  metal  tubes  of  sufficient 
uniformity  that  the  current  distribution,  and  hence  the  resistance, 
does  not  change  with  frequency. 

The  accuracy  of  ordinary  measurements  requires  that  the  re- 
sistance of  the  standard  be  constant  to  i  per  cent.  The  maximum 
size  of  wire  of  various  materials  that  may  be  used  can  be  found 
from  Table  1 8,  page  310,  for  the  highest  frequency  that  is  to  be 
used.  The  diameter  required  for  any  given  accuracy  and  a  given 
limiting  frequency  may  be  calculated  from  (207) ,  page  300 . 

These  fine  wires  which  must  be  used  as  high-frequency  standards 
will  not  carry  much  current  without  serious  heating.  They  must 
therefore  be  used  in  measurements  with  caution.  When  a  stand- 
ard is  required  to  carry  large  currents  it  can  not,  in  general,  be 
obtained  by  putting  several  fine  wires  in  parallel.  The  design  of 
such  a  resistance  standard  is,  in  fact,  identically  the  same  problem 
as  the  design  of  high-frequency  ammeters  for  large  currents. 
(Sec.  41,  p.  144.) 

The  ideal  resistance  standard  would  not  change  any  of  the  con- 
stants of  a  circuit  except  resistance  when  it  is  inserted  in  the 
circuits.  A  wire,  however,  has  some  inductance,  and  since  the 
inductances  used  in  radio  circuits  are  small  the  inductance  of 
a  wire  standard  of  resistance  can  not  be  neglected.  The  induc- 

35601°— 18 12 


1 76  Circular  of  the  Bureau  of  Standards 

tance  must  be  made  extremely  small  by  using  very  short  wires,  or 
its  effect  must  be  minimized,  by  substituting  for  the  resistance 
wire  a  copper  wire  of  the  same  length  whenever  the  resistance 
wire  [is  removed  from  the  circuit.  The  first  of  these  alterna- 
tives is  followed  in  the  use  of  a  short  slide  wire,  which  gives 
a  continuous  variation  of  resistance;  the  contact  with  such  a 
slide  wire  must,  of  course,  be  of  small  and  constant  resistance. 
The  second  alternative  is  followed  in  sets  of  resistance  standards. 
The  inductance  of  the  copper  wire  link  will  be  practically  the  same 
as  the  inductance  of  the  resistance  standard  substituted  for  it, 
and  the  slight  difference  between  them  will  not  ordinarily  affect 
the  total  inductance  of  the  circuit  provided  the  resistance  wire  and 
the  copper  link  are  short. 

A  set  of  resistance  standards  for  high  frequency  may,  therefore, 
be  a  set  of  short  wires  of  approximately  equal  lengths,  a  portion 
of  the  length  consisting  of  a  very  fine  wire.  The  wire  must  be  of 
very  small  diameter  to  obtain  fairly  large  resistances  in  the  short 
length  allowed.  This  fine  wire  is  of  resistance  material,  the  length 
is  adjusted  to  give  the  required  resistance,  and  the  remainder  of 
the  length  is  of  relatively  thick  copper  wire.  In  a  set  used  by  this 
Bureau,  the  resistance  material  is  manganin,  used  because  the 
resistance  does  not  change  appreciably  with  temperature.  In 
order  to  protect  the  delicate  wires  from  breakage  each  is  mounted 
in  a  glass  tube.  The  copper  ends  of  each  standard  are  amalga- 
mated for  insertion  in  small  mercury  cups;  the  amalgamation 
must  be  renewed  at  frequent  intervals.  They  are  7  cm  long;  the 
resistances  have  values  from  0.2  to  40  ohms;  the  resistance  wires 
have  lengths  from  0.5  to  6  cm,  and  diameters  from  0.03  to  0.12  mm. 
On  account  of  the  wires  being  of  such  small  diameter,  care  is  neces- 
sary to  avoid  using  currents  that  would  overheat  them.  The 
inductances  vary  from  0.15  microhenry  for  the  4O-ohm  standard 
to  0.08  microhenry  for  the  o-ohm  copper  link.  The  difference, 
0.07  microhenry,  is  negligible  except  in  rare  cases.  The  resistances 
have  been  found  to  remain  satisfactorily  constant  for  several 
years. 

Decade  resistance  boxes  are  also  useful  as  standards  when 
properly  made  and  used  with  caution.  The  resistance  units  must 
be  made  very  short  and  of  sufficiently  fine  wire.  When  such  a  set 
is  used  in  a  circuit  of  low  inductance,  variation  of  the  resistance 
setting  may  vary  the  inductance  of  the  circuit.  In  some  methods 
of  resistance  measurement  this  merely  requires  retuning  to 
resonance. 


Radio  Instruments  and  Measurements  177 

46.  METHODS  OF  MEASUREMENT 

The  methods  of  measuring  high-frequency  resistance  may  be 
roughly  classed  as:  (i)  Calorimeter  method,  (2)  substitution 
method,  (3)  resistance-variation  method,  and  (4)  reactance- 
variation  method. 

The  fourth  has  frequently  been  called  the  "decrement  method," 
but  it  is  primarily  a  method  of  measuring  resistance  rather  than 
decrement,  exactly  as  the  resistance-variation  method  is.  Thus, 
the  measurement  of  decrement  is  the  same  problem  as  the  measure- 
ment of  resistance.  When  applied  to  determine  the  decrement 
of  trains  of  waves,  radio-resistance  measurement  accomplishes 
something  similar  at  high  frequencies  to  what  is  done  at  low  fre- 
quencies by  wave  analysis.  Either  may  be  used  to  measure  the 
decrement  of  a  wave  under  certain  conditions  and,  in  fact,  the 
results  of  resistance  measurement  by  any  method  may  be  ex- 
pressed in  terms  of  decrement. 

All  four  methods  may  be  used  with  either  damped  or  undamped 
waves,  though  in  some  of  them  the  calculations  are  different  in 
the  two  cases.  They  are  all  deflection  methods,  in  the  sense  of  de- 
pending upon  the  deflections  of  some  form  of  high-frequency 
ammeter.  In  the  first  and  second,  however,  it  is  only  necessary 
to  adjust  two  deflections  to  approximate  equality,  while  in  the 
third  and  fourth  the  deflections  may  have  any  magnitude. 

47.  CALORIMETER  METHOD 

This  method  may  be  used  to  measure  the  resistance  either  of  a 
part  or  the  whole  of  a  circuit.  The  circuit  or  coil  or  other  appa- 
ratus whose  resistance  is  desired,  is  placed  in  some  form  of  calo- 
rimeter, which  may  be  a  simple  air  chamber,  an  oil  bath,  etc.  The 
current  is  measured  by  an  accurate  high-frequency  ammeter,  and 
the  resistance  R*  is  calculated  from  the  observed  current  / 


FIG.  126. — Method  of  measuring  resistance 
by  calorimeter  method 

and  the  power,  or  rate  of  heat  production,  P. 

P  =  RJ>  (77) 


178  Circular  of  the  Bureau  of  Standards 

While  P  might  be  measured  calorimetrically,  in  practice  it  is 
always  measured  electrically  by  an  auxiliary  observation  in  terms 
of  low-frequency  or  direct  current.  Thus,  it  is  only  necessary  to 
observe  the  temperature  of  the  calorimeter  in  any  arbitrary  units 
when  the  high-frequency  current  flows  after  the  temperature  has 
reached  a  final  steady  state,  and  then  cause  low-frequency  current 
to  flow  in  the  circuit,  adjusting  its  value  until  the  final  temperature 
becomes  the  same  as  before.  Denoting  by  the  subscript  o  the 
low-frequency  values 

P0  =  R<Jo2 

P      RJ2 


P/o 


ForP  =  P0,  R*=~-  (78) 

From  the  known  low-frequency  value  of  the  resistance,  therefore, 
and  the  observed  currents,  the  resistance  is  obtained. 

The  high  and  low  frequency  observations  are  sometimes  made 
simultaneously,  using  a  duplicate  of  the  apparatus  whose  resist- 
ance is  desired,  placed  in  another  calorimeter  as  nearly  identical 
with  the  first  as  possible.  High-frequency  current  is  passed 
through  one,  low  frequency  through  the  other,  and  the  calorim- 
eters kept  at  equal  temperatures  by  means  of  some  such  device  as 
a  differential  air  thermometer  or  differential  thermoelement.  To 
compensate  for  inequalities  in  the  two  sets  of  apparatus,  the  high 
and  low  frequency  currents  are  interchanged.  This  method  may 
be  found  more  convenient  in  some  circumstances,  but  the  extra 
complication  of  apparatus  is  usually  not  worth  while,  and  the 
value  of  the  measurement  depends  upon  the  accurate  observation 
of  the  high-frequency  current  /,  just  as  the  simpler  method  does. 

The  calorimeter  method,  while  capable  of  high  accuracy,  is 
slow  and  less  convenient  than  some  of  the  other  methods.  It 
has  been  used  by  a  number  of  experimenters  to  measure  the 
resistance  of  wires  and  coils. 

48.  SUBSTITUTION  METHOD 

This  method  is  applicable  only  to  a  portion  of  a  circuit.  Sup- 
pose that  in  Fig.  126  the  coil  L  is  loosely  coupled  to  a  source  of 
oscillations.  The  capacity  C  is  varied  until  resonance  is  obtained, 
and  the  current  in  the  ammeter  is  read.  A  resistance  standard  is 
then  substituted  for  the  apparatus  Rs.  and  varied  until  the  same 


Radio  Instruments  and  Measurements  1  79 

current  is  indicated  at  resonance.  If  the  substitution  has  changed 
the  total  inductance  or  capacity  of  the  circuit,  the  returning  to 
resonance  introduces  no  error  when  undamped  or  slightly  damped 
electromotive  force  is  supplied,  provided  the  change  of  condenser 
setting  introduces  either  a  negligible  or  known  resistance  change. 
In  the  case  of  a  rather  highly  damped  source,  however,  the 
method  can  only  be  used  when  the  resistance  substitution  does 
not  change  the  inductance  or  capacity  of  the  circuit.  The  un- 
known R*  is  equal  to  the  standard  resistance  inserted,  provided 
the  electromotive  force  acting  in  the  circuit  has  not  been  changed 
by  the  substitution  of  the  standard  for  R*  ;  this  condition  is  dis- 
cussed below. 

Unequal  Deflections.  —  The  resistance  standards  usually  used 
are  not  continuously  variable,  and  hence  the  standard  used  may 
give  a  deflection  of  the  ammeter  somewhat  different  from  original 
deflection. 

To  determine  the  resistance  in  this  case,  three  deflections  are 
required,  all  at  resonance.  The  apparatus,  say  an  absorbing  con- 
denser, of  unknown  resistance  Rx  is  inserted  and  the  current  /x 
observed;  then  a  similar  apparatus  of  known  resistance  RN  is 
substituted  for  it  and  the  current  /N  observed  ;  and  finally  a  known 
resistance  Rt  is  added  and  the  current  /t  observed.  The  relations 
between  these  quantities  and  the  electromotive  force  involve  the 
unknown  but  constant  resistance  of  the  remainder  of  the  circuit 
R,  thus,  assuming  undamped  emf, 


./N 


r-' 

from  which,  Rx-R*  =  R,  ~  -  (79) 


Or,  in  case  the  resistance  of  a  current-square  meter,  thermoele- 
ment, or  similar  apparatus  is  desired,  the  procedure  would  be  to 
observe  the  current  /  when  the  resistance  R  is  that  of  the  circuit 
alone,  the  current  7X  when  the  apparatus  of  resistance  Rx  is 
inserted  in  the  circuit,  and  the  current  It  when  a  known  resistance 
R,  is  substituted. 


i8o  Circular  of  the  Bureau  of  Standards 

Then 


From  which, 


Tfr 


This  method  is  closely  related  to  the  resistance-variation  method ; 
see  formula  (80)  below. 

Application  of  Method. — The  substitution  method  is  very  con- 
venient and  rapid,  and  is  suitable  for  measurements  upon  antennas, 
spark  gaps,  etc.,  and  for  rough  measurements  of  resistances  of 
condensers  and  coils.  In  radio  laboratory  work,  however,  using 
delicate  instruments  and  with  loose  coupling  to  the  source  of 
oscillations,  it  is  found  that  in  some  cases  it  is  not  a  highly  accurate 
method,  except  for  measuring  small  changes  in  resistance  of  a 
circuit.  The  reason  for  this  is  that  there  are  other  electromotive 
forces  acting  in  the  circuit  than  that  purposely  introduced  by  the 
coupling  coil,  viz,  emf's  electrostatically  induced  between  various 
parts  of  the  circuit.  When  the  apparatus  under  measurement  is 
removed  from  the  circuit,  these  emf's  are  changed,  and  there  is  no 
certainty  that  when  the  current  is  made  the  same  the  resistance  has 
its  former  value.  Something  of  the  same  difficulty  enters  into  the 
question  of  grounding  the  circuit  in  the  following  method,  as 
discussed  below. 

49.  RESISTANCE  VARIATION  METHOD 

As  this  method  measures  primarily  the  resistance  of  the  whole 
circuit,  the  principle  may  be  readily  understood  from  the  diagram 
of  the  simple  circuit,  Fig.  127. 

If  the  resistance  of  some  particular  piece  of  apparatus,  inserted 
at  P,  for  example,  is  to  be  found,  the  resistance  of  the  circuit  is 
measured  with  it  in  circuit  and  then  remeasured  in  the  same  way 
with  it  removed  or  replaced  by  a  similar  apparatus  of  known 
resistance;  and  the  resistance  of  the  apparatus  is  obtained  by 
simple  subtraction. 


Radio  Instruments  and  Measurements  181 

The  measurement  is  made  by  observing  the  current  /  in  the 
ammeter  A  when  the  resistance  R^  has  its  zero  or  minimum  value, 
then  inserting  some  resistance  Ri  and  observing  the  current  /j.' 
Let  R  denote  the  resistance  of  the  circuit  without  added  resist- 
ance, and  suppose  a  sine-wave  electromotive  force  E  introduced 
into  the  circuit  by  induction  in  the  coil  L  from  a  source  of 
undamped  waves.  For  the  condition  of  resonance 


r  E 

•I™ 


R+R, 
from  which  the  resistance  of  the  circuit  is  given  by 

(80) 


R  = 


/ 

7T1 


P  SJ 

o 

LO 


FIG  127. — Circuit  for  measuring  resistance 
by  the  resistance  variation  method 

The  same  method  can  be  employed  using  damped  instead  of 
continuous  waves,  and  can  even  be  used  when  the  current  is  sup- 
plied by  impulse  excitation,  but  the  equations  are  different;  see 
(89)  and  (90)  below. 

Precautions, — A  limitation  on  the  accuracy  of  the  measurement 
is  the  existence  of  the  emfs  electrostatically  induced  that  were 
mentioned  above.  In  the  deduction  of  (80)  it  is  assumed  that 
E  remains  constant.  The  virtue  of  this  method  is  that  these 
emfs  may  be  kept  substantially  constant  during  the  measure- 
ment of  resistance  of  the  circuit.  They  will  invariably  be  altered 
by  the  substitution  of  the  apparatus  whose  resistance  is  desired, 
but  the  resistance  of  the  circuit  is  measured  accurately  in  the  two 
cases  and  the  difference  of  the  two  measurements  gives  the  resist- 
ance sought.  In  order  to  keep  these  stray  electromotive  forces 
unchanged  when  Rt  is  in  and  when  it  is  out  of  circuit,  particular 
attention  must  be  paid  to  the  grounding  of  the  circuit.  The 
shield  of  the  condenser  and  the  ammeter  (particularly  if  it  is  a 


1  82  Circular  of  the  Bureau  of  Standards 

thermocouple  with  galvanometer)  have  considerable  capacity  to 
ground  and  are  near  ground  potential.  A  ground  wire,  if  used, 
must  be  connected  either  to  the  condenser  shield  or  to  one  side 
of  the  ammeter.  If  connected  to  the  high-potential  side  of  the 
inductance  soil  absurd  results  will  be  obtained.  The  resistance 
R!  also  must  be  inserted  at  a  place  of  low  potential,  preferably 
between  the  shielded  side  of  the  condenser  and  ammeter. 

Furthermore,  care  must  be  taken  that  the  coupling  between  the 
measuring  circuit  and  the  source  is  not  too  close.  Otherwise  the 
current  in  the  source  and  hence  the  emf  E  will  vary  somewhat 
when  RI  is  inserted.  This  will  give  incorrect  resistance  values 
which  will  depend  upon  the  magnitude  of  R^  Whether  such  an 
effect  is  present  can  be  judged  by  opening  and  closing  the  meas- 
uring circuit  and  no  ling  whether  this  produces  a  considerable 
change  in  the  ammeter  reading  in  the  source  circuit;  or  by  repeat- 
ing the  measurement  with  reduced  coupling.  In  order  that  the 
measurement  can  be  made  using  very  loose  coupling  it  is  necessary 
either  to  have  a  source  of  considerable  power  or  to  use  a  sensitive 
current-measuring  device  such  as  thermocouple  and  galvanometer. 

Use  of  Thermocouple.  —  As  regularly  carried  out  at  the  Bureau 
of  Standards,  in  the  resistance-variation  method  a  pliotron  is 
used  as  a  source  of  undamped  emf,  and  current  is  measured  with 
a  low-resistance  thermocouple  in  series  in  the  circuit.  The  cur- 
rents corresponding  to  given  deflections  of  the  thermocouple  gal- 
vanometer are  obtained  from  a  calibration  curve;  or  from  the 
law  Joe/2,  where  d  =  deflection,  if  the  instrument  follows  this  law 
sufficiently  closely.  When  the  deflections  follow  this  law,  equa- 
tion (80)  becomes 

T_      *i  (81) 


Several  values  of  resistance  Rt  are  usually  inserted  in  the  circuit 
and  the  corresponding  deflections  obtained;  the  resulting  values 
of  R  are  averaged.  An  example  is  given  on  page  192  below. 

When  the  thermocouple  follows  the  square  law  accurately,  the 
quarter  deflection  method  may  be  used,  which  eliminates  all  cal- 
culation. When  the  deflection  d^  is  %  d,  equation  (81)  becomes 

R  =  R!  (82) 

This  method  requires  a  variable  resistance  standard  such  that  7?t 
can  be  varied  continuously  in  order  to  make  dt  just  equal  to 


Radio  Instruments  and  Measurements  183 

%  d.  Practically  the  same  method  is  used  if  the  resistance  is 
varied  by  small  steps,  as  in  a  resistance  box,  and  interpolat- 
ing between  two  settings  of  R^  Suppose  that  the  two  values 
of  jRj  are  R2  and  R3  and  the  corresponding  deflections  d2  slightly 
less  than  %  d,  and  d3  slightly  greater  than  %  d, 

R=R3+(R2-R3)  (83) 


Use  of  Impulse  Excitation.  —  The  procedure  for  the  resistance- 
variation  method  is  the  same  when  the  current  is  damped  as 
when  undamped.  When  the  circuit  is  supplied  by  impulse  exci- 
tation, so  that  free  oscillations  are  produced,  the  theory  of  the 
measurement  is  very  simple.  The  current  being  /  when  the 
resistance  is  R  and  1^  when  the  resistance  R1  is  added,  the  power 
dissipated  in  the  circuit  must  be  the  same  in  the  two  cases,  because 
the  condenser  in  the  circuit  is  charged  to  the  same  voltage  by 
each  impulse  which  is  impressed  upon  it,  and  there  is  assumed 
to  be  no  current  in  the  primary  after  each  impulse. 
Therefore,  RI2  = 
whence, 


This  simple  deduction  of  this  equation  is  equivalent  to  the 
proof  of  (61)  on  page  93.  As  mentioned  there,  this  method  can 
be  regarded  as  a  true  measurement  of  decrement,  since  the  cur- 
rent flowing  in  the  circuit  has  the  natural  decrement  of  the  cir- 
cuit. It  is  difficult  to  obtain  high  accuracy  by  the  method  in 
practice  because  of  the  difficulty  of  obtaining  pure  impulse  exci- 
tation. An  example  of  such  a  measurement  is  given  on  page 
190  below. 

The  method  is  specially  convenient  when  an  instrument  is  used 
in  which  the  deflection  d  is  proportional  to  the  current  squared. 
Then  (84)  becomes 

R-R  (85) 


This  is  still  further  simplified  if  the  resistance  Rl  is  adjustable,  so 
that  dt  can  be  made  equal  to  one-half  d.  The  equation  then 
reduces  to 

R=R!  (86) 

This  is  commonly  known  as  the  half  -deflection  method. 


1  84  Circular  of  the  Bureau  of  Standards 

Use  of  Damped  Excitation.  —  The  resistance  variation  method  has 
already  been  shown  to  be  usable  with  either  undamped  or  free 
oscillations.  It  can  also  be  used  when  the  source  of  emf  is  damped 
so  that  both  forced  and  free  oscillations  exist  in  the  circuit. 
Equation  (62)  on  page  94  above  gives  the  relation  between  the 
currents  and  the  decrements  of  the  circuits.  As  there  stated, 
such  a  measurement  may  be  looked  upon  as  a  direct  measure- 
ment of  decrement.  It  is  possible  to  calculate  either  the  decre- 
ment 5'  of  the  supplied  emf  as  in  equations  (63)  to  (65)  when  the 
constants  of  the  measurement  circuit  are  known,  or  the  decre- 
ment 8  or  resistance  R  of  the  measuring  circuit  when  8'  is  known. 

A  convenient  form  of  the  solution  for  R  is  obtained  from  equa- 

D  r> 

tion  (66)  and  the  relations  8  =  TTJ  —  and  5,  =  TTJ-^, 

Lu  Lea 

KI  2 

(87) 


where 

*  =  '+yVa  (88) 

This  is,  of  course,  not  an  explicit  solution  for  R,  since  K  involves 
5  and  therefore  R,  but  gives  a  ready  means  of  finding  R  or  5  when 
the  sum  of  the  two  decrements  8'  +  8  is  known  from  some  other 
measurement,  such  as  the  reactance-variation  method  described 
below.  Thus,  a  combination  of  the  two  methods  gives  both  5' 
and  5,  or  5'  and  R. 

There  are  two  interesting  special  cases  in  which  the  measure- 
ment is  simplified.  When  the  decrement  8'  of  the  supplied  emf 
is  very  small  and  is  negligible  compared  with  5,  equation  (87) 
reduces  to 

#=  (89) 


identical  with  (80)  above,  the  equation  for  the  use  of  undamped 
emf.  This  is  to  be  expected,  since  undamped  emf  is  the  limiting 
case  of  small  decrement.  When,  on  the  other  hand,  5  and  8t  are 
both  very  small  compared  with  8',  K  becomes  unity  and  equation 
(87)  reduces  to 

R=RI-  (90) 


This  happens  to  be  the  same  as  equation  (84)  above,  the  equa- 
tion for  the  use  of  impulse  excitation.     The  proof  given  here 


Radio  Instruments  and  Measurements 


185 


can  not,  however,  be  regarded  as  a  deduction  of  equation  (84) 
for  impulse  excitation,  as  it  has  been  by  some  writers,  since 
Bjerknes'  equation  (p.  187)  is  involved,  which  assumes  that  8' 
and  5  are  both  small. 

50.  REACTANCE  VARIATION  METHOD 

This  has  been  called  the  decrement  method,  a  name  which  is 
no  more  applicable  to  this  than  to  the  other  methods  of  resistance 
measurement  since  all  measure  decrement  in  the  same  sense  that 
this  does.  That  the  method  primarily  measures  resistance  rather 
than  decrement  is  seen  from  the  fact  that  in  its  simple  and  most 
accurate  form  it  utilizes  undamped  current,  which  has  no  decre- 
ment. 


FlG.  128. — Circuit  for  measuring  resistance 
by  the  reactance  variation  method 

The  method  is  analogous  to  the  resistance-variation  method, 
two  observations  being  taken.  The  current  IT  in  the  ammeter 
is  measured  at  resonance,  the  reactance  is  then  varied  and  the 
new  current  7t  is  observed.  The  total  resistance  of  the  circuit  R 
is  calculated  from  these  two  observations.  The  reactance  may 
be  varied  by  changing  either  the  capacity,  the  inductance,  or  the 
frequency,  the  emf  being  maintained  constant.  The  reactance  is 
zero  at  resonance  and  it  is  changed  to  some  value  Xt  for  the  other 
observation.  With  undamped  emf  E,  the  currents  are  given  by 

E3 


From  these  it  follows  that 

r>       -y 

t\.  —  -^-i 


(91) 


This  has  a  similarity  to  R=Rt      l     ,   the   equation    (80)    for 

•*  ~A 
the  resistance-variation  method.     It  is  also  interesting  that  when 


1  86  Circular  of  the  Bureau  of  Standards 

the  reactance  is  varied  by  such  an  amount  as  to  make  the  quantity 
under  the  radical  sign  equal  to  unity,  the  equation  reduces  to 

£-*,  (92) 

This  is  similar  to  R=R19  which  is  the  equation  for  the  quarter- 
deflection  and  half-deflection  resistance  variation  methods. 

Special  Cases  of  Method.  —  When  the  reactance  is  varied  by 
changing  the  setting  of  a  variable  condenser,  the  equation  (91) 
becomes  (27)  given  on  page  38.  For  variation  of  the  inductance, 
(91)  becomes 

±-2  (93) 


For  variation  of  the  frequency,  (91)  becomes 


r 
-Y/T^  (94) 

It  must  be  noted  that  variation  of  the  frequency  requires  some 
alteration  in  the  source  of  emf  ,  and  the  greatest  care  is  necessary 
to  insure  that  the  condition  of  constant  emf  is  fulfilled. 

A  convenient  method  which  differs  slightly  from  those  just 
described  is  to  observe  two  values  of  the  reactance  both  corre- 
sponding to  the  same  current  7j  on  the  two  sides  of  the  resonant 
value  7r.  For  observation  in  this  manner  of  two  capacity  values 
Ci  and  C3,  _ 

i    C.-Q    /-jrT- 

- 


Decrement  Calculation.  —  It  is  often  convenient  to  calculate 
directly  the  decrement  of  the  circuit  instead  of  the  resistance. 
Formulas  for  decrement  exactly  corresponding  to  the  resistance 
formulas  already  given  are  obtained  by  application  of  the  simple 
relations  between  resistance  and  decrement  and  are  the  same  as 
formulas  (96)  to  (100)  below  with  5'  omitted.  The  formulas 
thus  obtained  are  rigorous,  as  are  the  foregoing  resistance  formu- 
las, for  undamped  emf,  and  hold  with  sufficient  accuracy  for 
damped  emf  when  the  damping  is  negligibly  small. 

When  the  damping  of  the  supplied  emf  is  appreciable,  the 
same  procedure  is  followed  in  making  the  measurement,  and  the 
equations  are  only  slightly  different.  When  the  emf  is  supplied 
by  coupling  to  a  primary  circuit  in  which  current  is  flowing  with 
a  decrement  8',  Bjerknes'  classical  proof  shows  that  the  sum  of 


Radio  Instruments  and  Measurements  187 

the  primary  and  secondary  decrements  is  given  by  the  same 
expression  as  that  which  gives  the  decrement  5  of  the  secondary 
when  the  emf  is  undamped.  Thus  (27),  (93),  (94),  and  (95) 
correspond  to 


rtjj  (98) 

; 
_.i  (99) 

Formula  (98)  is  also  equivalent  to 

^         /         JM 

l—r  (I0°) 


These  formulas  are  correct  only  when:  (i)  The  coupling  between 
the  two  circuits  is  so  loose  that  the  secondary  does  not  appre- 
ciably affect  the  primary,  (2)  5'  and  5  are  both  small  compared 

(C  —  C) 
with  2Tr,  and  (3)  the  ratio  -^-^ — -  and  the  corresponding  ratios 

are  small  compared  with  unity. 

In  any  of  these  methods  the  calculation  is  obviously  simplified 
if  the  reactance  is  varied  by  such  an  amount  as  to  make  If  =  ^2/r2. 
This  is  done  very  easily  when  the  current  measuring  instrument 
is  graduated  in  terms  of  current  squared.  The  quantity  under 
the  square-root  sign  in  all  the  preceding  equations  becomes  unity, 
greatly  simplifying  the  formulas.  A  still  further  simplification 
by  which  all  calculation  is  eliminated  is  utilized  in  special  decre- 
meters  as  described  below,  section  55. 

51.  RESISTANCE  OF  A  WAVE  METER 

The  accurate  measurement  of  resistance  or  decrement  of  a  wave- 
meter  circuit  is  of  first  importance  because  the  wave  meter  is  fre- 
quently used  to  measure  other  resistances  and  the  decrements  of 
waves.  It  is  the  calibration  of  a  resistance-measuring  standard. 
Several  forms  of  the  resistance-variation  and  the  reactance- 
variation  methods  may  be  used. 


1 88  Circular  of  the  Bureau  of  Standards 

The  resistance  of  a  wave  meter  is,  of  course,  not  a  single,  con- 
stant value.  It  varies  with  frequency  and  with  the  detecting  or 
other  apparatus  connected  to  the  wave-meter  circuit.  Usually 
both  the  resistance  and  the  decrement  of  the  circuit  vary  with  the 
condenser  setting.  It  is  usually  desirable  to  express  either  re- 
sistance or  decrement  in  the  form  of  curves  for  the  several  wave- 
meter  coils,  each  for  a  particular  detecting  apparatus  or  other 
condition.  An  example  of  such  a  curve  is  given  on  page  190. 

Resistance  Variation. — Any  of  the  forms  of  the  resistance- 
variation  method  may  be  used.  The  apparatus  and  procedure 
are  the  same  in  all  cases.  The  wave-meter  coil  is  loosely  coupled 
to  the  source.  The  current  is  read  on  the  indicating  device  shown 
schematically  as  A  in  Fig.  129.  A  resistance  standard  of  the  type 
already  described  is  then  inserted  at  R!  and  the  current  read 


FIG.  129. — Measurement  of  wave  meter  resistance 

again.  The  calculation  of  resistance  depends  on  the  damping  of 
the  source  and  the  kind  of  current-measuring  device. 

When  a  pliotron,  arc,  or  other  source  of  undamped  waves  is 
used,  formula  (80)  above  is  used.  When  the  current-measuring 
device  is  a  current-square  meter,  thermocouple  or  crystal  detector 
with  galvanometer,  or  other  apparatus  which  is  so  calibrated  that 
deflections  are  accurately  proportional  to  the  square  of  the  current, 
and  when  in  addition  a  continuously  variable  resistance  standard 
is  used,  the  quarter-deflection  method  may  be  employed  elimi- 
nating all  calculation.  As  explained  on  page  182,  the  resistance 
of  the  circuit  is  equal  to  the  inserted  resistance  required  to  reduce 
the  deflection  to  one-quarter. 

When  a  buzzer  or  other  damped  source  is  used,  some  auxiliary 
measurement  or  special  condition  is  needed,  in  order  to  evaluate 
or  eliminate  the  decrement  of  the  source.  If  this  decrement  is 
known,  the  decrement  or  resistance  of  the  wave-meter  circuit 
may  be  obtained.  The  solution  is,  however,  complicated  and, 
as  a  matter  of  fact,  this  method  in  not  used.  Instead  of  5', 
the  decrement  of  the  source,  being  known  explicitly,  the  more 
usual  case  is  that  8'  +6,  the  sum  of  the  decrements  of  source  and 


Radio  Instruments  and  Measurements  189 

wave  meter,  is  known  from  a  measurement  by  the  reactance-vari- 
ation method.     The  wave-meter  resistance  is  then  calculated  by 


and    resistance    and  decrement  are  related 


o  +o 

r> 

by  8  =  —  r  and  the  similar  formulas  given  on  page  316.     The  calcu- 

lation is  considerably  simplified  in  the  two  special  cases  of  df  very 
small  or  5  very  small;  formulas  (98)  and  (99),  respectively,  apply. 
The  latter  is  identical  with  the  equation  for  impulse  excitation, 
but  with  that  exception  these  methods  may  be  used  only  when 
5'  and  5  are  both  small. 

For  impulse  excitation  from  a  buzzer  or  other  source,  equation 
(84)  is  used  to  calculate  the  resistance.  When  the  current  indi- 
cator is  calibrated  in  terms  of  the  square  of  current  and  the 
resistance  standard  is  continuously  variable,  the  measurement  is 
conveniently  made  by  the  half-deflection  method.  In  this  case 
the  resistance  of  the  circuit  equals  the  inserted  resistance. 

Reactance  Variation.  —  This  method  may  be  used  with  either 
a  damped  or  undamped  source.  When  the  emf  is  undamped  or  of 
extremely  small  damping,  formulas  (27)  and  (93)  to  (95)  apply. 
It  is  customary  to  reduce  the  labor  of  computation  by  varying  the 


/    72 
reactance  by  such  an  amount  that  .*/       *       =i,  in  which 

»  1  r        ./i 
±(Cr-Q 


case 


r> 

taCfC 

R=±a(L-Lr)  (102) 

g_  ±L  (co2  — ay2)  (103) 

i      C2  — Cj  (104) 

.TV  =  —         — 7; — TT~ 
2CO        C2  L! 

When  damped  emf  is  used,  formulas  (96)  to  (100)  apply.     They 


also  are  simplified  in  practice  by  making  */  —£± : 

\  IT        *j 

They  require  either  that  5',  the  decrement  of  the  source,  be 
known,  or  that  another  relation  be  obtained  between  5'  and  5  by 
an  independent  measurement.  It  is  not  often  that  a  source 


190 


Circular  of  the  Bureau  of  Standards 


of  fixed,  known  decrement  is  maintained  in  a  laboratory,  as  the 
decrement  varies  with  frequency  and  every  other  condition  of  use. 
Hence,  the  usual  procedure  is  the  combination  of  this  measurement 
with  a  resistance-variation  measurement  as  described  above. 

An  example  of  measurement  of  wave-meter  resistance  expressed 
in  terms  of  decrement  is  given  in  Fig.  130.  This  shows  the  results 
of  two  independent  measurements,  one  by  the  resistance-variation 


004 


0.03 


OiOZ 


I 

•  =*  RESISTANCE  VARIATION 

(IMPULSE  EXCITATION) 

X—  REACTANCE  VARIATION 

CJMOAMPED  EXCITATION) 


20  <**  6O  SO  100  ;20  J4-O  t6O  ISO 

FIG.  130. — Variation  of  the  decrement  of  a  wave  meter  with  condenser  setting 


method,  using  impulse  excitation  and  equation  (84),  and  another 
by  the  reactance-variation  method  employing  equation  (27). 

52.  RESISTANCE  OF  A  CONDENSER 

The  methods  ordinarily  used  for  measurement  of  resistance  of  a 
condenser  or  of  an  inductance  coil  require  a  variable  condenser 
whose  effective  resistance  must  be  either  negligibly  small  or  accu- 
rately known.  This  condenser  is  used  to  retune  the  circuit  to 
resonance  after  the  unknown  is  taken  out  of  the  circuit,  The 
standard  condensers  of  negligible  resistance  used  at  the  Bureau 
of  Standards  are  described  on  page  119.  These  measurements 
may  be  made  with  an  ordinary  wave  meter,  provided  the  resist- 
ance of  the  circuit  is  accurately  known  for  different  condenser 
settings. 

Simple  Methods. — The  simplest  method  is  that  of  substitution. 
The  condenser  to  be  tested  is  connected  in  series  with  a  coil  and 
an  ammeter  of  some  sort,  and  loosely  coupled  to  a  source.  The 
condenser  is  then  replaced  by  the  standard  condenser  and  a  series 
resistance.  The  resistance  required  to  make  the  deflection  at 
resonance  the  same  as  before  is  taken  as  the  resistance  of  the 
condenser.  This  method  is  not  very  accurate,  because  the  change 
of  condensers  changes  the  emf's  electrostatically  induced  in  the 
circuit. 


Radio  Instruments  and  Measurements  191 

Another  method  utilizes  the  principle  of  reactance  variation. 
The  frequency  supplied  to  the  condenser  circuit  is  varied  by 
changing  the  setting  of  a  condenser  in  the  supply  circuit.  Under 
certain  conditions,  equation  (27)  or  (95)  applies.  The  method 
can  be  made  to  give  phase  differences  directly  by  use  of  a  suit- 
able scale  of  phase  differences  on  the  condenser  in  the  supply 
circuit.  (See  sec.  55.) 

Precision  Method. — Accurate  measurements  may  be  made  by 
the  resistance- variation  method.  The  circuit  is  tuned  to  resonance 
by  varying  the  frequency  supplied,  and  the  total  resistance  of  the 
circuit  is  measured,  with  the  unknown  condenser  in  circuit.  It  is 
replaced  by  the  standard  condenser,  the  setting  of  which  is  varied 
until  resonance  is  obtained,  and  then  the  resistance  of  the  circuit 
is  measured  again.  The  difference  of  the  two  measured  values  is 


QJ      v^   Source 
O 
R  O 

-WWWWSM 


FIG.  131. — Circuit  used  for  measurement  of 
high-frequency  resistance  of  a  condenser 

the  resistance  of  the  condenser  under  test.  As  previously  noted, 
precautions  are  necessary  to  avoid  changing  the  stray  electro- 
static emf's  in  the  circuit  when  the  resistance  Rl  is  introduced. 
In  respect  to  this  it  has  been  found  desirable  to  insert  Rt  between 
the  condenser  and  the  current-measuring  device,  and,  if  a  ground 
wire  is  used,  to  connect  it  to  the  shielded  side  of  the  condenser  or 
to  the  ammeter  case.  Also,  the  coupling  must  be  loose  enough 
so  that  too  much  power  is  not  withdrawn  from  the  source. 

The  manipulation  is  made  more  convenient  by  using  a  double- 
throw  switch  to  place  the  two  condensers  in  circuit.  The  base  of 
the  switch  must  be  a  material  which  has  very  small  phase  differ- 
ence; paraffin  has  been  found  suitable.  The  resistance  standards, 
when  in  the  shape  of  short  links,  may  be  used  as  part  of  this 
switch,  as  shown  at  R!  in  Fig.  132.  Another  refinement  of  the 
measurement  is  to  place  a  small  variable  condenser  C\  of  negligible 
resistance  in  parallel  with  the  inductance  coil.  This  gives  a  fine 
adjustment  to  resonance. 

Example. — An  example  of  a  measurement  at  one  frequency 
with  a  pliotron  as  a  source  of  undamped  emf  is  given  below.     The 

35601°— 18 13 


Circular  of  the  Bureau  of  Standards 


condenser   is   a   fixed   condenser   with   molded   dielectric.     The 
column  headed  "d"  gives  deflections  of  galvanometer  attached  to 


FIG.  132. — Circuit  for  precision  measurement  of  con- 
denser resistance  -with  switching  device  and  small 
tuning  condenser 

a  thermocouple;  deflections  are  proportional  to  the  square  of 
the  current. 

TABLE  4. — Observations  on  Resistance  of  a  Mica  Condenser 

[0-0.00406  M'.  L™ 40  nh,  X— 760  m,  /?x— ^?N=o.o9  ohm=  resistance  of  condenser  X;  phase  difference™ 

0.09(4,060) 
6-s ITT -3  -J 


c 

Ri 

Galvanometer 

Vd    . 

R         R' 

Zero 

Deflection 

d 

Mean  d 

ar1 

/d-l 
Vdi 

X  
N  

0 
0.503 
.810 
1.042 
.810 
.503 
0 
0 
.503 
.810 
1.042 
.810 
.503 
0 

13.95 
13.95 
•13.92 
13.90 
13.95 
13.90 
13.92 
13.98 
13.92 
13.92 
13.88 
13.88 
13.90 
13.95 

45.70 
31.68 
27.30 
24.82 
27.18 
31.75 
45.95 
48.58 
32.45 
27.72 
25.15 
27.70 
32.60 
48.50 

31.75 
17.73 
13.38 
10.92 
13.23 
17.83 
32.03 
34.60 
18.53 
13.80 
11.27 
13.82 
18.70 
34.55 

31.89 
17.78 
13.30 
10.92 

1.  480] 
1.479ll.477 
1.472 

1.390 
1.391  1.38s 
1.387 

0.340 
.548 
.708 

34.58 
18.62 
13.81 
11.27 

.362 
.582 
.752 

Radio  Instruments  and  Measurements 


193 


The  resistance  of  a  condenser  is  generally  measured  at  several 
frequencies.  If  the  resistance  is  mainly  due  to  dielectric  absorp- 
tion, the  resistance  is  generally  inversely  proportional  to  fre- 
quency. Variable  condensers  are  usually  measured  at  several 
settings.  For  a  variable  air  condenser  with  semicircular  plates 
having  a  small  resistance  mainly  due  to  dielectric  absorption  in 
the  separating  insulators,  the  resistance  is  inversely  proportional 
to  the  square  of  the  setting. 

53.  RESISTANCE  OF  A  COIL 

The  resistance  of  a  coil  depends  upon  its  position  in  the  circuit, 
i.  e.,  whether  the  emf  acting  upon  the  circuit  is  impressed  in  the 
coil  itself  or  at  some  other  point  in  the  circuit.  This  difference 
arises  from  the  effects  due  to  the  capacity  of  the  coil.  When  the 
emf  is  induced  in  the  coil  itself,  the  capacity  of  the  coil  is  to  be 
considered  in  series  with  the  inductance  of  the  coil  but  in  parallel 


— vwwwwv-0 — ' 


FIG.  133. — Circuit  for  measuring  high-fre- 
quency resistance  of  a  coil  by  the  substi- 
tution method 

with  the  rest  of  the  circuit.  When,  however,  the  emf  is  impressed 
at  some  other  point  in  the  circuit,  the  coil  capacity  and  inductance 
are  in  parallel  with  each  other.  When  the  resistance  of  a  coil  is 
measured  by  any  of  the  methods  given  in  this  section,  the  value 
of  the  resistance  obtained  is  valid  for  the  coil  only  when  used  in 
the  same  position  relative  to  the  driving  emf. 

The  simplest  method  is  that  of  substitution.  In  a  wave  meter 
circuit  coupled  to  a  source  of  emf  by  an  independent  coupling 
coil,  the  deflection  is  first  observed  with  the  coil  whose  resistance 
is  desired  in  circuit.  If  the  wave  meter  condenser  resistance 
is  known  for  various  settings,  the  coil  is  then  replaced  by  a  stand- 
ard coil  whose  resistance  is  known,  the  condenser  retuned  to 
resonance,  and  resistance  inserted  until  the  deflection  is  the 


1 94  Circular  of  the  Bureau  of  Standards 

same  as  before.  The  change  of  coils  may  be  made  by  a  double- 
throw  switch.  If  a  variable  inductor  of  known  resistance  is 
available,  the  procedure  is  still  simpler,  as  the  condenser  setting 
need  not  be  changed;  the  coil  under  test  is  replaced  by  the  variable 
inductor,  which  is  used  to  obtain  resonance,  and  then  resistance  is 
inserted  to  equalize  deflections.  As  in  previous  cases,  the  sub- 
stitution method  is  not  an  accurate  one,  but  is  valuable  for 
speedy  determination  of  approximate  values. 

More  accurate  measurements  may  be  made  by  either  the  resist- 
ance-variation or  the  reactance-variation  method.  Either  of 
these  methods  may  be  used  to  determine  the  resistance  in  each 
of  the  three  following  procedures. 

Known  Circuit. — The  circuit  consists  of  the  unknown  coil,  a 
condenser,  the  current-indicating  instrument,  and  connecting  leads. 
The  emf  is  introduced  into  the  circuit  by  coupling  to  the  unknown 

coil.  The  resistance  of  the  total 
circuit  is  obtained  by  the  resist- 
ance-variation or  the  reactance- 
variation  method  and  the  coil 
resistance  determined  by  sub- 
tracting the  resistance  of  the  rest 

FIG.  134. — Coil  resistance  measurement  in  ^ 

terms  of  known  circuit  or  standard  coil        of     the     Circuit.    ^  The     Condenser 

should  be  practically  perfect  or 

of  known  resistance,  and  the  leads  should  be  of  negligible  or 
calculable  resistance.  If  the  indicating  instrument  is  a  thermo- 
element or  current-square  meter  with  fine  wire  heating  element, 
its  resistance  may  be  determined  with  direct  current.  In  doubtful 
cases  it  can  be  determined  at  the  frequency  of  the  measurement 
by  a  separate  experiment  as  outlined  on  page  179.  Thermo- 
elements of  low  resistance,  as  described  on  page  157,  are  especially 
suited  for  this  measurement,  since  the  precision  is  higher  if  the 
resistance  of  the  indicating  device  is  a  small  part  of  the  total 
resistance  of  the  circuit. 

Known  Coil. — Two  resistance  measurements  at  high  frequency 
are  required  when  the  unknown  coil  is  compared  with  a  standard 
coil  of  known  resistance.  The  inductance  of  the  standard  coil 
should  be  of  the  same  order  of  magnitude  as  that  of  the  unknown, 
but  need  not  be  equal  to  it. 

The  standard  is  substituted  for  the  unknown  and  the  condenser 
varied  for  resonance  and  the  resistance  of  the  circuit  obtained  in 
each  case. 

Auxiliary  Coil. — The  third  procedure  is  to  measure  the  resistance 
of  a  simple  radio  circuit  consisting  of  a  condenser,  ammeter, 


Radio  Instruments  and  Measurements  195 

and  any  coil,  all  in  series,  and  then  insert  the  coil  to  be  tested, 
re  tune  to  resonance,  and  measure  the  resistance  again.  The  coil 
must  be  inserted  at  such  a  place  in  the  circuit  that  it  has  no 
mutual  inductance  with  the  auxiliary  coil,  and  so  that  there  is  no 
emf  induced  in  it  by  the  source. 

54.  DECREMENT  OF  A  WAVE 

Any  measurement  of  the  resistance  of  a  circuit  by  the  methods 
already  given  is  in  a  sense  a  measurement  of  decrement,  since  it 
enables  calculations  of  the  decrement  which  the  circuit  would 
have  when  oscillating  freely.  In  particular,  the  use  of  impulse 
excitation  with  the  resistance-variation  method  measures  the 
decrement  of  the  oscillations  actually  existing  in  the  measuring 
circuit  during  the  measurement,  and  therefore  the  decrement  of  the 
wave  emitted  by  the  measuring  circuit. 

There  is  a  class  of  decrement  measurements  entirely  apart  from 
the  measurement  of  decrement  or  resistance  of  a  circuit.  This  is 
the  determination  of  decrement  of  an  emf,  due  either  to  a  nearby 
antenna  or  other  circuit  or  to  a  wave  traveling  through  space. 
The  fundamental  principles  of  decrement  measurement  have 
been  given  in  section  26  above.  A  simple  wave-meter  circuit  is 
placed  so  as  to  receive  the  wave  and  a  decrement  measurement  is 
made  by  either  the  resistance-variation  or  the  reactance-variation 
method.  If  the  resistance  of  the  wave-meter  circuit  is  known 
for  the  frequency  and  other  conditions  of  the  measurement,  the 
decrement  of  the  wave  is  calculated.  For  the  resistance-variation 
method  8',  the  decrement  of  the  wave,  is  given  by  formula  (63), 
page  63,  or  one  of  the  simplified  formulas  (64)  or  (65) . 

For  the  reactance-variation  method  the  decrement  is  given  by 
(96)  to  (100),  page  187,  or,  in  case  the  simplified  procedure  is 
followed,  making  I?  =  yZIr2, 

y+g-x±(c^"Q  (105) 

g'+s=7r±(j;-Lr)  (I06) 

L.T 

'-•)  (I07) 

(108) 


196  Circular  of  the  Bureau  of  Standards 

The  measurement  may  also  be  made  by  a  direct-reading  decre- 
meter  (see  next  section),  and  (5  '  +8)  read  directly  from  the  scale 
of  the  instrument. 

When  the  resistance  or  decrement  of  the  wave-meter  circuit  is 
not  known,  measurements  are  made  by  both  methods:  and  the 
combination  of  the  two  yields  the  value  of  df,  making  use  of  equa- 
tion (63). 

55.  THE  DECREMETER 

A  decremeter  is  a  wave-meter  conveniently  arranged  for  meas- 
urements of  resistance  or  decrement.  The  forms  usually  employed 
make  use  of  the  reactance-variation  method.  Ways  have  been 
devised  for  manipulating  the  instrument  in  such  a  way  that  the 
decrement  may  be  read  directly  from  a  scale.  While,  of  course, 
resistance  can  be  calculated  from  a  measured  value  of  decrement, 
the  principle  application  of  the  decremeter  is  in  the  measurement 
of  the  decrement  of  a  wave. 

Another  important  use  is  in  the  measurement  of  phase  differ- 
ence of  a  condenser.  Since  the  decrement  due  to  a  condenser  is 
TT  times  its  phase  difference,  a  measurement  of  decrement  gives 
directly  the  phase  difference;  if  desired,  the  scale  may  be  cali- 
brated in  terms  of  phase  difference  instead  of  decrement. 

Determination  of  the  Scale  of  a  Decremeter.  —  Any  wave  meter 
whose  circuit  includes  some  form  of  ammeter  may  be  fitted  with 
a  special  scale  from  which  decrements  may  be  read  directly. 
The  procedure  for  a  wave  meter  having  any  sort  of  variable  con- 
denser is  given  here. 

In  the  usual  use  of  the  reactance  variation  method  of  deter- 
mining decrement,  the  current  IT  is  observed  when  the  condenser 
is  adjusted  to  the  value  Cr  to  produce  resonance,  and  the  con- 
denser is  then  changed  to  another  value  C  and  the  current  /t 
read.  When  the  second  condenser  setting  is  such  that  the  If  = 

—  7r2,   the  decrement  is  calculated  by 

<-+»-,      -Q 


A  certain  value  of  decrement  therefore  corresponds  to  that  dis- 
placement of  the  condenser's  moving  plates  which  varies  the 
capacity  by  the  amount  (Cr-C).  The  displacement  for  a  given 
decrement  will  in  general  be  different  for  different  values  of  C, 
the  total  capacity  in  the  circuit.  At  each  point  of  the  con- 
denser scale,  therefore,  any  displacement  of  the  moving  plates 

which  changes  the  square  of  current  from  7r2  to  -  7r2  means  a 
certain  value  of  (8  ' 


Radio  Instruments  and  Measurements  197 

A  special  scale  may  therefore  be  attached  to  any  condenser 
with  graduations  upon  it  and  so  marked  that  the  difference  be- 

tween the  two  settings,  when  the  square  of  current  is  7r2  and  -  7r2, 

is  equal  to  the  decrement.  The  spacing  of  the  graduations  at 
different  parts  of  the  scale  depends  upon  the  relation  beween 
capacity  and  displacement  of  the  moving  plates.  When  this  rela- 
tion is  known,  the  decrement  scale  can  be  predetermined.  A 
scale  may  therefore  be  fitted  to  any  condenser,  from  which  de- 
crement may  be  read  directly,  provided  the  capacity  of  the  circuit 
is  known  for  all  settings  of  the  condenser.  The  decrement  scale 
may  be  attached  either  to  the  moving-plate  system  or  to  the  fixed 
condenser  top.  It  is  usually  convenient  to  attach  it  to  the  unused 
half  of  the  dial  opposite  the  capacity  scale.  The  value  of  decre- 
ment determined  by  this  method  is  (df  +  6)  ,  where  6  is  the  decre- 
ment of  the  instrument  itself.  This  must  be  known  from  the 
calibration  of  the  instrument  (e.  g.,  as  in  sec.  51),  the  value  of  5', 
the  decrement  of  the  wave  under  measurement,  being  then  ob- 
tained by  subtraction. 

Simple  Direct-Reading  Decremeter.  —  It  is  particularly  easy  to 
make  a  decremeter  out  of  a  circuit  having  a  condenser  with  semi- 
circular plates.  Such  condensers  follow  closely  the  linear  law, 


where  6  is  the  angle  of  rotation  of  the  moving  plates  and  a  and 
C0  are  constants.  It  can  be  shown  that  the  decrement  scale  ap- 
plicable to  such  a  condenser  is  one  in  which  the  graduations  vary 
as  the  logarithm  of  the  angle  of  rotation.  Furthermore,  the  same 
decrement  scale  applies  to  all  condensers  of  this  type.  This  scale 
has  been  calculated  and  is  given  in  Fig.  135.  It  is  calculated  to 
fit  equation  (108),  for  observations  on  both  sides  of  resonance, 
and  not  for  equation  (105).  This  scale  may  be  used  on  any  con- 
denser with  semicircular  plates.  The  scale  may  be  cut  out  and 
trimmed  at  such  a  radius  as  to  fit  the  dial  and  then  affixed  to  the 
condenser  with  its  O  point  in  coincidence  with  the  graduation 
which  corresponds  to  maximum  capacity.  This  usually  puts  it  on 
the  unused  half  of  the  dial  opposite  the  capacity  scale.  If  the 
figures  are  trimmed  off,  they  can  be  added  over  the  lines  in  red 
ink.  If  it  is  desired  not  to  mutilate  the  page,  the  scale  may  be 
copied.  This  scale  gives  accurate  results  if  the  capacity  scale  is 
so  set  that  its  indications  are  proportional  to  the  capacity  in  the 
circuit. 


198 


Circular  of  the  Bureau  of  Standards 


A  measurement  of  decrement  is  made  by  first  observing  the 
current-square  at  resonance,  then  reading  the  decrement  scale  at 
a  setting  on  each  side  of  resonance  for  which  the  current-square 


FlG.  135. — Direct-reading  decrement  scale  for  any  semicircular- 
plate  condenser 

is  one-half  its  value  at  resonance.     The  difference  between  the 
two  readings  on  the  decrement  scale  is  the  value  of  8 '  +  8. 

The  scale  permits  accurate  measurement  of  fairly  large  decre- 
ments, but  offers  no  precision  in  +he  measurement  of  very  small 
decrements,  particularly  at  the  low-capacity  end  of  the  scale. 


Radio  Instruments  and  Measurements  199 

A  similar  scale  is  readily  made  to  read  phase  differences  di- 
rectly. The  readings  of  Fig.  135  are  all  multiplied  by  18.24,  t° 
give  phase  differences  in  degrees.  The  instrument  is  then  specially 
valuable  in  measuring  the  phase  differences  of  condensers. 

Decremeter  with  Uniform  Decrement  Scale. — Just  as  it  is  pos- 
sible to  determine  a  decrement  scale  to  fit  a  condenser  having  any 
sort  of  law  of  capacity  variation,  it  is  equally  possible  to  design 
a  condenser  with  capacity  varying  in  such  a  way  as  to  fit  any 
specified  decrement  scale.  A  uniform  decrement  scale — i.  e.,  one 
in  which  the  graduations  are  equally  spaced — is  particularly  con- 
venient, and  is  the  kind  used  in  the  Kolster  decremeter.  A  uni- 
form decrement  scale  requires  that  the  condenser  plates  be  so 
shaped  that  for  any  small  variation  of  setting  the  ratio  of  the 
change  in  the  capacity  to  the  total  capacity  is  constant.  The 
condenser  required  to  give  this  uniform  scale  has  its  moving 
plates  so  shaped  that  the  logarithm  of  the  capacity  is  proportional 
to  the  angle  of  rotation  of  the  plates.  This  condenser  is  dis- 
cussed above  on  page  116. 

The  decremeter  is  fully  described  in  Bulletin  of  the  Bureau  of 
Standards,  11,  page  421 ;  1914  (Scientific  Paper  No.  235).  A  view 
of  the  inside  arrangement  is  shown  in  Fig.  218,  facing  page  320,  of 
this  circular.  The  spiral  shape  of  the  condenser  moving  plates  is 
shown.  Figures  219  and  220,  facing  page  32,  show  different  forms 
in  which  the  instrument  is  made.  The  decrement  scale  is  not 
attached  directly  to  the  moving  plates,  but  is  on  a  separate  shaft 
geared  to  the  moving  plates  at  a  6  to  i  ratio.  The  decrement  scale 
is  thus  opened  out,  so  that  very  precise  measurenemts  may  be 
made. 

To  measure  decrement  the  condenser  setting  is  first  varied  to 
obtain  maximum  deflection  of  the  current-square  meter,  and  then 
varied  until  half  this  deflection  is  obtained.  The  movable  decre- 
ment scale  is  then  set  at  zero  and  clamped  to  its  shaft,  and  the 
condenser  setting  is  varied  until  the  same  deflection  is  obtained 
on  the  other  side  of  the  maximum.  The  reading  on  the  decrement 
scale  is  the  value  of  decrement  sought. 

This  decremeter  is  used  in  the  inspection  service  of  the  Bureau 
of  Navigation,  Department  of  Commerce,  and  by  radio  engineers 
in  the  Army  and  Navy  and  elsewhere. 


2OO  Circular  of  the  Bureau  of  Standards 

SOURCES  OF  HIGH-FREQUENCY  CURRENT 

56.  ELECTRON  TUBES 

For  the  purposes  of  measurement  as  well  as  in  the  transmission 
of  radiograms,  sources  which  furnish  undamped  currents  are 
coming  into  more  general  use  than  those  which  supply  damped 
oscillations.  Thus  the  various  forms  of  spark,  used  almost 
exclusively  in  the  past,  are  giving  way  to  the  electron-tube  gen- 
erator, the  arc,  and  the  high-frequency  alternator.  These  sources 
are  used  both  in  radio  communication  and  in  laboratory  work, 
but  for  the  latter  the  electron  tube  is  preeminently  the  best. 
When  the  source  furnishes  undamped  current  many  methods  of 
measurement  are  simplified,  and  since  the  sharpness  of  tuning  is 
increased,  a  higher  precision  is  obtainable  in  methods  which 
depend  upon  tuning  to  resonance. 

Since  most  of  the  measurements  at  radio  frequencies  are  based 
upon  deflection  methods,  the  primary  requirement  of  a  source 
for  such  measurements  is  that  the  intensity  and  frequency  of  the 
generated  current  be  constant.  It  is  only  within  the  past  few 
years  that  an  almost  ideal  source  has  become  available,  viz,  elec- 
tron-tube generators,  such  as  the  pliotron,  audion,  oscillion,  etc. 
These  electron  tubes  consist  of  an  evacuated  bulb  containing  the 
following  three  elements: 

(a)  A  heated  filiament  which  acts  as  a  source  of  electrons. 

(&)  A  metal  ' '  plate ' '  placed  near  the  electron  source.  (Across 
the  plate  and  filament,  outside  of  the  bulb,  is  connected  a  bat- 
tery, so  that  an  electron  current  flows  from  filament  to  plate.) 

(c)  A  "grid"  consisting  of  fine  wire  or  of  a  perforated  metal 
sheet  placed  between  plate  and  filament  so  that  the  electrons 
have  to  pass  through  the  grid  to  get  from  filament  to  plate. 

Using  these  sources,  undamped  high-frequency  current  can  be 
obtained  which  is  as  steady  as  the  current  from  a  storage  battery 
and  strictly  constant  in  frequency.  Furthermore,  these  genera- 
tors are  extremely  flexible  as  to  the  frequencies  which  can  be 
obtained,  the  same  tubes  having  been  used  to  generate  currents 
ranging  in  frequency  from  i  cycle  in  two  seconds  to  50000000 
cycles  per  second.25  A  high-frequency  output  of  500  watts  or 
more  has  been  attained  with  a  single  tube.  For  ordinary  meas- 
uring purposes  with  sensitive  indicating  devices,  such  as  a  thermo- 
element and  galvanometer,  about  5  watts  of  high-frequency 

25  See  reference  No.  138,  Appendix  2. 


Radio  Instruments  and  Measurements  201 

output  are  sufficient.  When  it  is  desired  to  use  low-range  hot- 
wire ammeters,  10  to  20  watts  may  be  required. 

On  account  of  the  extreme  importance  of  the  three-electrode 
tubes,  both  as  generators  and  as  detectors,  and  since  the  full 
realization  of  their  utility  and  a  satisfactory  explanation  of  their 
functioning  are  of  recent  date,  it  is  worth  while  to  outline  rather 
fully  the  phenomena  upon  which  their  operation  is  explained. 

Thermionic  Emission. — The  modern  conception  of  current  flow 
in  metals  assumes  that  the  conduction  of  electricity  consists  in 
the  motion  of  electrons  (see  p.  8)  under  the  action  of  an  applied 
electromotive  force.  When  not  acted  upon  by  an  external  emf 
these  small  negatively  charged  particles  move  about  in  the  metal 
in  zigzag  paths  in  all  directions,  colliding  with  the  atoms  of  the 
metal.  Their  mean  velocity  of  motion  depends  upon  the  tem- 
perature, increasing  with  the  temperature.  At  the  surface  of  the 
metal,  according  to  the  theory  of  Richardson,26  the  electrons  are 
restrained  from  leaving  the  metal  by  electric  forces  entirely  simi- 
lar to  the  molecular  forces  which  cause  the  surface  tension  of  a 
liquid.  Further,  just  as  in  the  evaporation  of  a  liquid,  a  certain 
number  of  electrons  will  in  each  second  attain  a  high  enough 
velocity  to  escape  from  the  metal,  and  since  the  mean  velocity 
increases  with  the  temperature  the  number  of  electrons  escaping 
per  second  will  increase  with  the  temperature.  The  heated  fila- 
ment in  the  audion  or  pliotron  is  in  this  manner  a  source  of 
electrons.  The  withdrawal  of  the  negative  electrons  from  the 
heated  filament  leaves  it  positively  charged,  thus  tending  to  draw 
them  back  again  and  a  state  of  equilibrium  may  be  attained  in 
which  the  same  number  are  drawn  back  per  second  as  are  being 
emitted.  When,  however,  a  body  maintained  at  a  positive 
potential  relative  to  the  filament  is  brought  into  the  field  a  certain 
proportion  of  the  electrons  will  be  attracted  to  the  positive  body 
and  constitute  a  current  of  electricity  between  the  filament  and 
the  positively  charged  body.  In  the  electron  tube  this  body  is 
the  "plate"  and  its  potential  is  maintained  positive  with  respect 
to  the  filament  by  a  battery  commonly  called  the  B  battery.  If 
in  the  case  of  a  tube  with  an  extremely  high  vacuum  the  voltage 
of  the  B  battery  is  increased,  the  flow  of  electrons  or  "plate 
current"  will  increase  up  to  a  point  where  practically  all  of  the 
electrons  emitted  by  the  filament  are  being  drawn  over  to  the 
plate.  If,  on  the  other  hand,  the  plate  voltage  is  kept  constant 
and  the  filament  temperature  increased,  thus  increasing  the 

28  See  reference  No.  131,  Appendix  2. 


202 


Circular  of  the  Bureau  of  Standards 


number  of  electrons  emitted  per  second,  the  plate  current  will 
also  increase  up  to  a  certain  temperature,  but  beyond  this  tempera- 
ture will  remain  practically  constant  even  though  more  electrons 
are  being  given  off.  The  explanation  of  this  behavior 27  is  that 
the  stream  of  negative  electrons  flowing  through  the  tube  acts  as 
a  space  charge  of  negative  electricity  which  neutralizes  the  field 
due  to  the  positive  plate.  In  consequence  only  a  limited  number  of 
electrons  can  flow  to  the  plate  per  second  with  a  given  plate  voltage, 
and  the  remainder  are  compelled  to  return  to  the  filament  again. 

Grid  Control. — If  in  any  way  this  space  charge  is  neutralized, 
there  will  be  an  increase  in  the  plate  current;  on  the  other  hand, 


Grid   Voltage 

FlG.  136. — Variation  of  plate  current  (usu- 
ally in  milliamperes)  and  grid  current  (us- 
ually in  microamperes}  with  grid  -voltage 

anything  that  will  aid  the  space  charge  will  result  in  a  decrease 
in  the  plate  current.  In  the  audion  or  pliotron  these  effects  are 
brought  about  by  the  grid  of  wires  between  the  plate  and  filament. 
If  this  grid  is  charged  positively  with  respect  to  the  filament,  the 
effect  of  the  space  charge  will  be  neutralized  to  an  extent  depend- 
ing upon  the  charge  on  the  grid,  and  the  electron  current  through 
the  tube  will  increase  until  the  field  due  to  the  grid  charge  is  also 
neutralized  by  the  space  charge.28  Some  few  electrons  will  strike  the 

27  See  reference  No.  133,  Appendix  2. 

28  In  tubes  which  are  not  evacuated  to  a  high  degree,  the  residual  gas  may  become  ionized  and  markedly 
affect  the  behavior  of  the  tube.     The  ionization  of  the  gas  tends  to  neutralize  the  space  charge,  thus  per- 
mitting larger  currents  to  pass  through  the  tube.    To  a  certain  extent  such  ionization  is  of  value  in  the  use 
of  the  tube  as  a  detector,  though  when  the  ionization  becomes  intense  and  the  tube  shows  a  blue  glow,  so 
large  a  current  passes  through  the  tube  that  it  is  unaffected  by  variations  of  the  grid  voltage  and  its  detect- 
ing qualities  are  lost. 


Radio  Instruments  and  Measurements 


203 


grid  wires  and  there  will  result  a  flow  of  current  in  the  grid  circuit, 
but  this  will,  in  general,  be  small  relative  to  the  plate  current.  If 
the  grid  is  charged  negatively  with  respect  to  the  filament,  the 
charge  on  the  grid  will  then  aid  the  space  charge  in  driving  the 
electrons  back  to  the  filament,  resulting  in  a  lowering  of  the  plate 
current.  In  this  latter  case  the  number  of  electrons  striking  the 
grid  will  be  very  small  and  in  consequence  practically  no  current 
will  flow  in  the  grid  circuit.  The  control  of  the  plate  current  by 
the  grid  voltage  and  also  the  dependence  of  the  current  in  the  grid 
circuit  upon  the  grid  voltage  are  shown  in  curves  of  Fig.  136. 
Curve  A  shows  the  current  in  the  plate  circuit  when  the  B  battery 
is  kept  constant,  but  different  voltages  are  applied  between  the 
grid  and  that  terminal  of  the  filament  to  which  the  negative  of  the 
filament  battery  is  connected.  Curve  B  represents  on  a  magnified 


FIG.  137. — Scheme  of  connections  for  determining  characteristic  curves 

scale  the  current  in  the  grid  circuit  for  different  voltages  of  the 
grid  with  respect  to  the  negative  terminal  of  the  filament.  The 
ordinates  and  shape  of  these  so-called  characteristic  curves  depend 
upon  a  number  of  factors,  such  as  B  battery  voltage,  fineness  and 
spacing  of  the  grid  wires,  location  of  the  grid  relative  to  the  other 
elements,  etc.  Fig.  137  shows  the  scheme  of  connections  which 
may  be  used  in  determining  such  curves.  The  ammeter  Al 
measures  the  current  in  the  grid-filament  circuit  and  A2  measures 
the  plate  current.  By  means  of  the  sliding  contact  on  the  shunt 
resistance  to  the  battery  C,  the  voltage  between  the  filament  and 
grid  may  be  varied  and  made  positive  or  negative,  the  voltage 
being  read  by  the  voltmeter  V.  The  B  battery  voltage  is  held 
constant  while  the  curve  is  taken. 


204  Circular  of  the  Bureau  of  Standards 

57.  ELECTRON  TUBE  AS  DETECTOR  AND  AMPLIFIER 

As  Detector  of  Damped  Oscillations. — A  single  tube  may  per- 
form separately  or  simultaneously  the  functions  of  a  detector, 
amplifier,  and  generator.  It  will  first  be  considered  as  a  simple 
detector  of  damped  oscillations.  The  circuits  shown  in  Fig.  138 
indicate  one  possible  way  of  using  the  tube  as  a  detector.  The 
circuit  LC\  is  tuned  to  the  oscillations  in  the  antenna  A.  The  C 
battery  with  variable  resistance  permits  the  adjustment  of  the 
grid  potential  with  respect  to  the  filament,  so  that  the  tube  may  be 
worked  at  any  point  on  the  characteristic  curve  of  the  plate  cur- 
rent. Suppose  that  this  voltage  is  adjusted  to  correspond  to  the 


FIG.  138. — Possible  circuits  for  using  the  electron  tube  as  a  detector  of  damped 

oscillations 

point  X  (Fig.  1 39)  where  the  change  in  slope  of  the  curve  is  large. 
If  now  a  train  of  oscillations  is  set  up  in  the  antenna  and  hence  in 
the  secondary  circuit,  the  alternating  voltage  across  the  condenser 
terminals  will  be  superimposed  upon  the  steady  voltage  of  the 
C  battery.  It  will  be  seen  from  the  characteristic  curve  that  an 
increase  in  voltage,  say,  from  a  to  b,  produces  a  large  increase  in 
the  plate  current  (i.  e.,  from  x  to  y) ,  while  a  decrease  in  voltage  of 
the  same  amount  from  a  to  c  produces  a  much  smaller  change 
(from  x  to  z)  in  the  current.  Thus,  as  the  result  of  a  wave  train 
such  as  (i)  in  Fig.  140,  the  plate  current  will  be  changed  about  its 
normal  value  in  some  such  way  as  (2)  which  is  equivalent  to  a 
resultant  increase  in  plate  current.  This  increase  of  plate  current 


Radio  Instruments  and  Measurements 


205 


during  a  train  of  waves  gives  rise  to  a  pulse  of  current  in  the  tele- 
phone as  shown  in  (3).  This  pulse  will  act  upon  the  telephone 
diaphragm,  and  if  the  wave  trains  and  hence  the  pulses  in  the 
telephone  current  are  arriving  at  the  rate  of  1000  per  second  (cor- 
responding to  the  spark  frequency  at  the  transmitting  station), 
a  looo-cycle  note  will  be  heard  in  the  phone.  This  use  of  the  tube 
as  a  detector  is  entirely  similar  to  the  use  of  a  crystal  detector. 

Condenser  in  Grid  Lead. — If  a  condenser  is  inserted  instead  of  the 
C  battery  in  the  lead  to  the  grid,  as  C2  in  Fig.  141,  the  behavior 


cab  O 

Grid  Voltaqe, 

FIG.  139. — Plate  characteristic,  showing  region  of  curve  where 
rectifying  action  is  large 

of  the  tube  as  a  detector  of  damped  29  oscillations  is  altered  and 
depends  to  a  great  extent  upon  the  characteristic  curve  of  the 
grid  current.  The  grid  is  insulated  from  the  filament  by  the  con- 
denser C2,  excepting  for  such  leakage  as  may  take  place  through 
this  condenser  or  in  or  about  the  tube.  Suppose  first  that  the 
tube  is  put  into  operation  with  the  grid  and  filament  at  the  same 
potential  and  with  no  incoming  oscillations.  It  will  be  seen  from 

29  Although  damped  oscillations  are  referred  to  here  and  in  the  usual  treatments  of  the  subject,  the 
same  considerations  apply  to  undamped  oscillations  which  are  periodically  interrupted  either  in  the  trans- 
mitting or  receiving  circuits  so  that  the  tube  receives  groups  or  trains  of  waves. 


206 


Circular  of  the  Bureau  of  Standards 


the  filament-grid  curve  that  there  will  be  a  flow  of  electrons  to  the 
grid  and  the  grid  will  become  negative  with  respect  to  the  filament, 


FlG.  140. — Action  of  the  electron  tube  as  a  detector:  (7)  Incoming  oscil- 
lations, (2)  variations  in  plate  current,  (j)  effective  telephone  pulses 

thereby  reducing  the  flow  to  itself  and  to  the  plate  until  the  leakage 
away  from  the  grid  is  equal  to  the  flow  to  it.     In  some  tubes  the 


FIG.  141. — The  electron  tube  as  a  detector  of  damped  oscillations,  using  a  con- 
denser in  the  grid  circuit 

grid  may  be  so  highly  insulated  that  it  accumulates  a  negative 
charge  sufficiently  high  to  reduce  the  plate  current  practically  to 


Radio  Instruments  and  Measurements 


207 


zero.     In  such  cases  it  is  necessary  to  provide  an  artificial  leak 
through  a  high  resistance  across  C2. 

Suppose  now  that  the  grid  has  attained  its  equilibrium  potential 
and  the  plate  current  its  corresponding  value  and  a  series  of  wave 
trains  impinges  upon  the  antenna  as  in  (i)  of  Fig.  142.  The 
oscillations  in  the  circuit  LC\  will  cause  the  grid  potential  to 
oscillate  about  its  normal  value.  When  the  grid  becomes  positive 


© 


Incom'mq     Oscillations 


J 


Grid   Potential 


4- 


Plate    Current 


Pulses  m   phone 

FIG.  142. — Action  of  the  electron  tube  as  a  detector  connected  as  in  Fig.  141 

relative  to  its  normal  value  there  will  be  a  considerable  increase  in 
the  flow  of  electrons  to  it,  overbalancing  the  reduction  in  the 
flow  when  on  the  negative  half  of  the  wave.  Thus,  during  a 
wave  train  the  grid  will  accumulate  a  negative  charge  and  its 
mean  potential  will  be  lowered,  as  in  (2)  of  Fig.  142.  In  conse- 
quence the  mean  plate  current  will  be  reduced.  However, 
between  wave  trains  the  excess  charge  on  the  grid  will  leak  off, 
restoring  the  plate  current  to  its  normal  value.  This  is  shown 

35601°— 18 14 


208 


Circular  of  the  Bureau  of  Standards 


in  (3)  of  Fig.  142.  Each  wave  train  will  produce  a  reduction 
in  the  current  through  the  phones  as  in  (4)  of  the  same  figure 
and  a  note  corresponding  to  the  wave  train  frequency  will  be 
heard. 


FiG.  143. — Use  of  electron  tube  as  an  amplifier 

Amplification. — If,  as  in  Fig.  143,  a  source  of  alternating  emf 
were  interposed  between  the  filament  and  grid  of  aji  audion 
or  pliotron,  the  potential  of  the  grid  with  respect  to  the  filament 
would  alternate  in  accordance  with  the  alternations  of  the 


FIG.  144. — Variations  of  plate  current  with  grid  -voltage 

generator.  These  variations  of  the  grid  potential  produce  changes 
in  the  plate  current  corresponding  to  the  plate  characteristic. 
If  the  mean  potential  of  the  grid  and  the  amplitude  of  its  alter- 
nations are  such  that  the  plate  current  is  always  in  that  portion  of 


Radio  Instruments  and  Measurements 


209 


its  characteristic  where  it  is  a  straight  line,  then  the  alternations 
of  the  grid  potential  will  be  exactly  duplicated  in  the  variations 
of  the  plate  current  and  the  latter  will  be  in  phase  with  the  former, 
at  least  in  a  high  vacuum  tube.  Thus,  if  (a)  of  Fig.  144  represents 
the  alternating  potential  of  the  grid,  then  (6)  would  represent  the 
fluctuations  of  the  plate  current.  For  a  given  amplitude  in  (a), 
the  amplitude  of  the  alternating  component  in  (b)  will  depend 
upon  the  steepness  of  the  plate  characteristic,  increasing  with 
increasing  slope.  The  alternator  in  the  grid  lead  supplies  only  the 
very  small  grid-filament  current,  thus  the  power  drawn  from  it  is 
extremely  small.  The  power  represented  by  the  alternating 
component  of  the  plate  current  is,  however,  considerable;  thus 
there  is  a  very  large  power  amplification.  This  larger  source  of 


5     P 

FIG.  145. — Use  of  electron  tube  as  a  regenerative  amplifier 

power  might  be  utilized  by  inserting  the  primary  P  of  a  trans- 
former in  the  plate  circuit,  as  in  Fig.  143,  in  which  case  the  alter- 
nating component  alone  would  be  present  in  the  secondary  5. 
This  illustrates  the  principle  of  a  vacuum  tube  as  a  relay.  The 
voltage  in  5  might  again  be  inserted  in  the  grid  lead  of  a  second 
vacuum  tube  and  with  proper  design  a  further  amplification  ob- 
tained in  the  plate  circuit  of  the  second  tube.  This  may  be 
carried  through  further  stages  and  illustrates  the  principle  of 
multiple  amplification. 

Regenerative    Amplification. — It    has    been    shown    by    E.    H. 
Armstrong80  that    amplification   similar   to   that   obtained  with 

*°  See  reference  No.  134,  Appendix  a. 


2IO 


Circular  of  the  Bureau  of  Standards 


several  stages  may  be  secured  with  a  single  tube.  Instead  of 
feeding  the  voltage  of  the  secondary  coil  5  into  the  grid  circuit  of  a 
second  tube  it  is  fed  back  into  the  grid  circuit  of  the  same  tube 
so  as  to  increase  the  voltage  operating  upon  the  grid.  This  results 
in  an  increased  amplitude  of  the  plate-current  alternations  which 
likewise  being  fed  back  into  the  grid  circuit  increases  the  voltage 
operating  upon  the  grid,  etc. 

One  form  of  the  so-called  feed-back  circuit  for  rectifying  and 
amplifying  damped  oscillations  is  shown  in  Fig.  145.  The  oper- 
ation of  the  circuit,  used  as  a  receiving  device,  is  the  same  as  that 
described  above  for  the  case  of  a  condenser  in  the  grid  lead.  The 
condenser  C2  is  merely  to  provide  a  path  of  low  impedance  across 
the  phones  for  the  high-frequency  oscillations.  The  coils  P  and  5 
constitute  the  feed  back  by  means  of  which  the  oscillations  in 


FIG.  146. — Use  of  electron  tube  as  a  generator 

the  tuned  circuit  are  reinforced.  The  mutual  inductance  between 
S  and  P  must  be  of  the  proper  sign  so  that  the  emf  fed  back  aids 
the  oscillations  instead  of  opposing  them. 

58.  ELECTRON  TUBE  AS  GENERATOR 

Generation  of  Oscillations. — If  the  coupling  between  the  coils 
P  and  5  in  Fig.  145  is  continuously  increased  and  the  values  of 
L,  S,  and  C  and  the  resistance  of  this  circuit  are  suitable  within 
certain  limits,  the  emf  fed  back  by  the  coil  P  into  the  oscillatory 
circuit  at  any  instant  will  become  greater  than  that  required  to 
just  sustain  the  oscillations  in  the  circuit.  In  this  case  any  oscil- 
lation, however  small  in  the  circuit  L,  S,  C,  will  be  continuously 
built  up  in  amplitude  until  a  limit  determined  by  the  character- 
istics of  tube  and  circuits  is  reached.  In  other  words,  the  tube 
self-generates  alternating  current  of  a  frequency  determined  by 
the  natural  frequency  of  the  oscillatory  circuit. 


Radio  Instruments  and  Measurements 


211 


Numerous  circuits  have  been  devised  to  produce  oscillations. 
Fig.  146  shows  a  method  of  connection  which  is  suitable  for  pro- 
ducing large  currents.  The  oscillatory  circuit  is  in  the  filament- 
plate  circuit  and  a  coil  between  filament  and  grid.  The  operation 
of  this  circuit  is  somewhat  different  from  that  outlined  above. 
Instead  of  transferring  all  of  the  energy  necessary  to  sustain  the 
oscillations  from  the  plate  to  the  grid  circuit  as  in  the  preceding 
case,  only  an  emf  which  serves  as  a  control  is  here  transferred. 
Thus,  the  grid  circuit  plays  a  similar  part  to  that  of  the  slide  valve 
in  a  reciprocating  engine.  The  path  of  the  current  flow  within 


FIG.  147. — Generating  circuits  in  which  the  oscillatory  circuit  is 
inductively  coupled  to  both  the  grid  and  plate  circuits 

the  tube  from  plate  to  filament  may  be  regarded  as  a  variable 
resistance,  the  value  of  which  depends  upon  the  potential  of  the 
grid.  If  the  potential  of  the  grid  is  alternating,  the  resistance 
will  increase  and  decrease  in  accordance,  thus  throwing  an  alter- 
nating emf  upon  the  oscillatory  circuit  in  series  with  this  resistance. 
The  oscillatory  circuit  which  determines  the  frequency  may  be 
a  separate  circuit,  as  in  Fig.  147.  Here  the  coupling  M2  supplies 
the  emf  to  reinforce  the  oscillations  and  Mx  furnishes  the  emf  to 
the  grid.  The  condenser  Q  is  a  large  fixed  condenser  which 
serves  as  a  path  of  low  impedance  across  the  battery  for  the  high- 
frequency  alternations  in  the  plate  circuit. 


212 


Circular  of  the  Bureau  of  Standards 


In  addition  to  the  above  types  of  circuit  in  which  electromag- 
netic coupling  between  the  plate  and  grid  circuits  is  used  to  trans- 
fer emfs  from  one  to  the  other,  there  are  also  circuits  in  which 
electrostatic  coupling  is  utilized.  This  is  illustrated  in  Fig.  148, 
in  which  the  condenser  C2  serves  as  the  coupling.  The  induc- 
tances L!  and  L2  should  be  variable  and  approximately  equal. 
Cf  is  a  fixed  condenser  which  serves  as  a  path  of  small  impedance 
for  the  high  frequency  around  the  battery.  The  frequency  is 
primarily  determined  by  the  inductances  Lx  and  L2  and  the  con- 
denser C2.  The  parallel  connection  of  C\  and  Lt  serves  as  an 
"absorbing"  circuit — that  is,  as  Cl  is  increased  from  a  very  low 


Ca 


FIG.  148. — Generating  circuits  in  which  the  plate  and  grid  circuits  are 
electrostatically  couphd 

value — the  current  circulating  around  this  circuit  will  increase  up 
to  a  certain  point  and  may  considerably  exceed  the  current  in  the 
other  portions  of  the  circuit. 

Reception  of  Undamped  Oscillations. — If  two  sources,  which 
separately  furnish  undamped  oscillations  of,  say,  100  ooo  and 
101  ooo  frequency,  as  shown  in  (a)  and  (6)  of  Fig.  149,  act  together 
upon  the  same  circuit,  the  resultant  oscillations  in  the  circuit, 
obtained  by  adding  the  components,  will  be  of  the  form  shown  in 
(c).  The  mode  of  adding  the  components  is  illustrated  in  Fig. 
150.  The  amplitude  of  the  combined  oscillation  will  rise  and  fall, 


Radio  Instruments  and  Measurements 


213 


becoming  a  maximum  when  the  component  oscillations  are  in 
phase  and  a  minimum  when  they  are  180°  out  of  phase.  The 
beats  or  periodic  rise  and  fall  in  amplitude  occur  at  a  rate  equal 
to  the  difference  in  frequencies  of  the  two  oscillations.  Thus,  the 


FIG.  149. — Principle  of  heterodyne  reception;  (a)  incoming 
oscillations,  (6)  oscillations  produced  by  the  tube,  (c)  result- 
ant current 

beat  frequency  in  the  case  assumed  above  would  be  101  ooo  — 
loo  000  =  1000  per  second.  If  rectified,  these  beats  will  produce 
a  note  in  a  telephone  of  like  frequency.  In  the  reception  of 
undamped  signals  by  this  method,  called  the  heterodyne  method, 


FIG.  150. — Mode  of  addiitg  component  oscillations 

the  incoming  signals  represent  one  component  oscillation.  The 
other  oscillation  is  generated  in  the  receiving  apparatus  and  both 
act  in  the  same  circuit.  The  rectified  resultant  furnishes  a 
musical  note  in  the  phones,  the  pitch  of  which  can  readily  be 


2I4 


Circular  of  the  Bureau  of  Standards 


altered  by  varying  the  frequency  of  the  local  source  of  oscilla- 
tions. The  electron  tube  may  serve  as  a  convenient  source  of 
local  oscillations  and  at  the  same  time  as  an  amplifier  and  detector 
of  the  received  signals.  This  is  called  the  autodyne  method. 
Numerous  circuits  may  be  utilized  to  produce  these  results,  of 
which  that  shown  in  Fig.  145,  page  209,  may  serve  as  an  illustra- 
tion. Incoming  signals  set  up  oscillations  in  the  antenna.  By 
means  of  the  coupling  between  the  antenna  and  coil  L  oscillations 


FlG.  151. — Variations  of  mean  grid  voltage  and  mean  plate  current  as 
beat  oscillations  are  being  produced 

of  the  same  frequency  are  set  up  in  the  circuit  L  C,  and  as  ex- 
plained above  are  amplified  on  account  of  the  feed  back  between 
S  and  P.  Further,  the  coupling  between  5  and  P  is  such  that 
the  tube  oscillates,  the  frequency  of  these  oscillations  depending 
largely  upon  the  constants  of  the  circuit  L  C.  If  this  latter  fre- 
quency is  adjusted  to  be  slightly  different  from  that  of  the  incom- 
ing oscillations,  beats  will  result  and  the  potential  of  the  grid  will 
follow  the  beat  oscillations.  Just  as  explained  before  in  the  case 


Radio  Instruments  and  Measurements 


215 


of  reception  with  a  grid  condenser,  there  will  be  an  increased  flow 
of  negative  electricity  from  the  filament  to  the  grid  when  this 
latter  is  positive  and  its  mean  potential  will  be  lowered.  Thus, 
as  the  oscillations  in  the  beat  are  increasing  the  potential  of  the 
grid  will  become  lower.  The  plate  current  will  follow  the  varia- 
tions in  potential  of  the  grid,  reproducing  the  beat  oscillations 
and  decreasing  in  mean  value  as  the  mean  potential  of  the  grid  is 
lowered.  The  curve  (a)  of  Fig.  151  represents  the  beat  oscillations 
in  the  circuit  L  C.  In  (6)  is  shown  the  oscillations  of  the  grid 
potential,  the  mean  potential  being  indicated  by  a  dotted  line. 


FlG.  152. — "  Ultraudion"  circuit  for  receiving  undamped  oscillations 

In  (c)  is  shown  the  plate  current,  the  mean  value  of  which  is  also 
shown  by  a  dotted  line.  The  telephone  current  will  likewise  cor- 
respond to  this  mean  value  and  hence  the  note  will  correspond  to 
the  beat  frequency.  In  Fig.  152  is  shown  the  connections  for  the 
circuit  used  by  L.  De  Forest,  the  inventor  of  the  audion,  for  the 
reception  of  undamped  oscillations  and  called  the  "  ultraudion. " 
The  oscillatory  circuit  is  connected  between  the  grid  and  plate 
with  a  condenser  in  the  grid  lead.  The  variable  condenser  C, 
shunted  across  the  plate  battery  and  phones  is  important  in  the 
production  of  oscillations;  in  general,  its  value  can  not  be  increased 
beyond  a  certain  point  without  stopping  the  oscillations. 


2l6 


Circular  of  the  Bureau  of  Standards 


By  this  beat  method  high  sensitiveness  and  selectivity  are 
attained  in  receiving.  Interference  is  minimized  because  even 
slight  differences  in  frequency  of  the  waves  from  other  sources 
result  in  notes  either  of  different  pitch  or  completely  inaudible. 

Tone  Modulation  of  Radio  Currents  from  Electron  Tube. — In 
undamped  wave  radio  transmitters,  the  radio-frequency  currents 
may  be  modulated  by  the  use  of  what  may  be  called  "tone  cir- 
cuits." It  is  then  possible  to  take  advantage  of  the  very  selective 
tuning  obtainable  with  undamped  wraves  without  employing  beat 
methods  of  reception.  Since  undamped  high-frequency  currents 
can  be  produced  from  electron  tubes,  it  is  particularly  convenient 
to  apply  to  these  tubes  devices  for  impressing  an  audible  tone  on 

the  currents  generated.     This   means 
>'  \  that  a  periodic  variation  of  the  ampli- 

tude of  the  radio-frequency  current  is 
produced,  this  periodic  variation  being 
of  audible  frequency.  Radio  currents 
modulated  in  this  way  may  be  pro- 
duced from  electron  tubes  in  the  three 
ways  described  below. 

Modulation  of  Electron  Stream. — The 
electron  current  through  the  tube  may 
be  modified  by  placing  the  tube  in  a 
strong  magnetic  field  which  varies  in 
strength  with  an  audible  period.  The 
circuits  shown  in  Fig.  146  or  in  Fig. 
1 60  may  be  used.  A  coil  wound  around 
the  electron  tube  is  supplied  with  cur- 
rent from  a  5oo-cycle  generator,  as  in 
Fig.  153,  or  a  direct  current  through  the 
coil  may  be  interrupted  by  means  of  a  buzzer. 

Modulation  of  Grid  Potential. — Instead  of  modulating  the 
current  by  external  means,  advantage  may  be  taken  of  the 
characteristics  of  the  tube  itself.  The  potential  of  the  grid  with 
respect  to  the  filament  may  be  varied  with  a  relatively  slow 
period  by  means  of  the  arrangement  shown  in  Fig.  154,  where  an 
audio-frequency  circuit  is  inserted  in  the  lead  to  the  grid  of  the 
electron  tube.  The  circuit  L3C3  is  tuned  to  resonance  with  the 
L2  circuit  which  is  coupled  loosely  to  it.  Any  of  the  various 
methods  of  generating  alternating  currents  may  be  used  for  this 
purpose  if  a  circuit  such  as  L3C3  is  inserted  in  the  grid  lead  and  an 
alternating  current  of  an  audible  frequency  induced  in  it.  For 


FIG.  153. — Modulation  of  gener- 
ated current  by  action  of  low- 
frequency  magnetic  field  on 
electron  stream 


Radio  Instruments  and  Measurements 


217 


example,  the  L2  circuit  may  be  supplied  from  a  5oocycle  genera- 
tor, with  L3  equal  to  50  millihenries  and  C3  equal  to  2  microfarads. 
The  audio-frequency  current  may,   like  the  radio-frequency 
current,  be  generated  by  an  electron  tube.     In  this  case  the  L, 


FIG.  154. — Modulation  of  generated  current  by  means  of  periodic  changes 
of  grid  voltage 

circuit  referred  to  above  is  replaced  by  the  oscillatory  circuit  of 
the  audio-frequency  generator  as  in  Fig.  155.  It  will  be  noted 
that  the  same  type  of  circuit  is  used  for  generating  audio  as  for 
generating  radio  currents,  it  being  necessary  merely  to  provide 


FlG.  155. — Method  of  using  an  electron  tube  for  producing  periodic  changes  of  grid  voltage 
in  another  tube  generating  radio-frequency  current 

suitable  values  of  L4,  L3  and  C2.  There  must  be  mutual  induc- 
tance between  L4  and  L2  of  Fig.  155  just  as  between  the  coils  in  the 
grid  and  plate  circuits  of  Fig.  146.  The  audio  and  the  radio 
frequency  generators  may  be  operated  from  the  same  batteries. 


218 


Circular  of  the  Bureau  of  Standards 


Self -Modulating  Tube. — In  the  methods  previously  described, 
means  for  modulating  are  provided  outside  of  the  radio-frequency 
tube.  This,  however,  is  not  necessary,  for  it  is  possible  to  generate 


FlG.  1 56. — A  rrangementfor  producing  modulated  radio-frequency 
current  by  use  of  a  single  tube 

both  audio  and  radio  frequency  currents  simultaneously  from  the 
same  tube.  Two  arrangements  of  circuits  whereby  this  may  be 
done  are  shown  in  Figs.  156  and  157.  In  these  two  diagrams  the 


FIG.  157. — Method  of  coupling  tone-circuit  generator  to  an  antenna 

radio-frequency  circuit  is  Lf^  The  circuits  L2C2  and  L3C9 
are  of  audio-frequency.  There  is  mutual  inductance  between 
L8  and  L,  as  well  as  between  L0  and  Lj. 


Radio  Instruments  and  Measurements  219 

Any  of  the  arrangements  described  above  may  be  used  to  produce 
modulated  radio-frequency  currents  in  an  antenna  by  coupling 
the  antenna  to  the  radio-frequency  coil  Ll  as  in  Fig.  157.  For 
making  signals  a  key  may  be  inserted  in  the  lead  to  the  grid  or  in 
the  connection  between  the  filament  and  the  B  battery,  or  in 
the  audio  circuit  L3C3  as  shown  in  Fig.  1 57. 

The  pitch  of  the  note  given  by  these  tone  transmitters  may  be 
varied  at  will  by  changing  the  constants  of  the  audio  circuits 
L3C3  and  L3C2.  Thus,  several  transmitters  may  operate  using 
the  same  wave  length  but  having  different  modulating  tones. 
The  receiver,  if  provided  with  means  for  tuning  to  the  audio  as 
well  as  to  the  radio  frequency,  will  be  free  from  interference 
even  by  other  stations  using  the  same  wave  length.  This  method 
of  transmitting  offers  the  considerable  advantage  that  the  tone 
is  a  pure  musical  note  and  does  not  change  in  pitch  with  slight 
changes  in  the  tuning  of  the  receiving  station  as  in  beat  meth- 
ods of  reception. 

Electron  Tube  as  Generator  for  Measurement  Purposes. — It  is 
desirable  in  generating  oscillations  for  measurement  purposes  that 
the  amplitude  and  frequency  of  the  generated  current  shall  be 
constant  and  that  the  set-up  shall  be  simple  and  flexible.  By 
the  latter  term  is  meant  that  a  wide  range  of  wave  lengths  may 
be  obtained  with  the  same  apparatus. 

Constancy  of  amplitude  and  frequency  are  easily  obtained. 
The  main  requirement  being  steadiness  in  the  batteries  supplying 
the  filament  heating  current  and  the  electron  current  between 
plate  and  filament.  High-frequency  current,  constant  both  in 
magnitude  and  in  frequency  to  better  than  one-tenth  of  i  per 
cent  over  long  intervals  of  time,  is  readily  obtained.  When  two 
or  more  tubes  are  operated  in  parallel  on  the  same  B  battery 
changes  occur  in  the  intensity  of  the  current  furnished  by  one 
tube  at  the  instant  the  second  tube  is  put  into  operation  or  when 
the  operation  of  the  second  tube  is  changed.  Independent  fila- 
ment batteries  should  always  be  used.  The  circuit  shown  in 
Fig.  146  is  simple  and  flexible.  The  frequency  generated  is  ordi- 
narily varied  by  changing  the  capacity  C\.  With  given  coils,  as 
the  capacity  is  increased,  there  comes  a  point  where  the  oscilla- 
tory current  falls  off  and  finally  "breaks".  It  is  then  necessary 
to  use  coils  of  greater  inductance  in  order  to  obtain  longer  wave 
lengths. 

Another  circuit  similar  to  the  above  and  which  has  shown 
itself  to  be  convenient  is  shown  in  Fig.  160.  Here  the  coils  Lx 


22O 


Circular  of  the  Bureau  of  Standards 


and  L2  may  be  wound  in  a  single  layer  adjacent  to  each  other 
on  the  same  form.  Taps  may  be  brought  out  on  each  coil  so  as 
to  use  the  number  of  turns  desired.  The  condensers  C2  and  Cs 
are  large  fixed-value  condensers  which  should  be  of  low  resistance. 
Q  is  the  tuning  condenser.  A  tungsten  lamp  is  introduced  in 
series  with  the  B  battery  to  protect  the  filament  of  the  tube  in 
case  of  an  accident.  The  measuring  circuit  may  be  coupled 
directly  to  the  coils  Llt  L2,  or  to  a  special  coil  of  a  few  turns 
inserted  in  series  with  either  of  these  coils,  preferably  on  the  side 
connected  to  the  B  battery  since  this  point  is  held  at  constant 
*  potential  by  the  large  capacity  of  battery  to  ground. 

The  B  battery  may  be  inserted  directly  in  the  lead  from  the 
plate  instead  of  adjacent  to  the  filament  as  shown  above.     With 


Plate, 


FIG.   160. — Scheme  of.  connections  for 
pliotron  generator 

such  connection,  however,  care  must  be  taken  that  there  is  very 
little  capacity  between  the  two  batteries  or  their  leads;  if  the 
batteries  or  their  leads  are  not  well  separated  and  insulated  from 
each  other,  the  high-frequency  current  is  much  reduced.  An 
advantage  of  locating  the  B  battery  adjacent  to  the  plate  is  that 
a  single  continuous  coil  may  provide  all  the  inductances  required 
in  the  circuits.  Thus,  as  shown  in  Fig.  161  below,  connections 
may  be  made  to  the  coil  LL  from  filament,  grid,  plate,  condenser, 
and  high-frequency  ammeter  by  movable  contacts.  Great  lati- 
tude of  adjustment  of  the  several  inductances  is  thus  allowed, 
and  the  connections  are  very  simply  shifted  from  one  type  of 
circuit  to  another,  so  that  the  proper  connections  to  give  maxi- 
mum current  for  any  wave  length  are  made  by  simply  sliding 


Radio  Instruments  and  Measurements 


221 


these  contacts.  An  advantage  of  the  mode  of  drawing  the  cir- 
cuits shown  in  Fig.  161  is  that  it  brings  out  that  the  several 
types  of  connection  are  equivalent. 

59.  POULSEN  ARC 

Another  valuable  source  of  undamped  oscillations  for  measure- 
ments with  moderate  or  high  power  is  the  Poulsen  arc.  If,  as 
in  Fig.  162,  an  ordinary  direct-current  carbon  arc  in  air  is  shunted 
by  a  circuit  containing  capacity  and  inductance  in  series,  oscilla- 
tions may  be  obtained  in  the  shunt  circuit.  Since  the  oscillations 
obtained  with  this  simple  arc  are,  in  general,  of  audible  frequency, 
the  arrangement  is  called  the  singing  arc.  Numerous  attempts 


FIG.  161. — Pliotron  generator  using  a  single  coil  with  sliding  contacts 

have  been  made  to  utilize  the  arc  in  air  as  a  generator  of  high- 
frequency  currents,  but  it  was  found  that  the  power  of  the 
oscillations  rapidly  decreased  with  increasing  frequency  so  that 
it  was  impossible  to  attain  frequencies  higher  than  about  10  ooo. 
V.  Poulsen,  however,  found  that  by  modifying  the  arc  in  the 
following  respects,  high  powers  could  be  obtained  at  least  up  to 
moderately  high  radio  frequencies: 

1 .  The  arc  is  surrounded  by  a  hydrocarbon  atmosphere  such  as 
coal  gas  or  alcohol  vapor. 

2.  Copper  instead  of   carbon   is   substituted  for   the  positive 
electrode. 

Further,  it  is  desirable  to  cool  the  copper  electrode  by  water 
circulation,  to  rotate  the  carbon  electrode,  and  (particularly  for 
high  powers)  to  provide  a  transverse  magnetic  field  across  the  arc 


222  Circular  of  the  Bureau  of  Standards 

to  blow  it  out.  The  source  supplies  several  hundred  volts.  The 
action  of  the  arc  in  generating  oscillations  is  roughly  the  following : 
When  no  current  is  flowing  through  the  gap  a  high  voltage  is  re- 
quired to  start  the  arc.  Immediately,  however,  upon  starting  the 
arc  the  path  of  the  discharge  is  ionized  and  the  resistance  of  the 
arc  is  greatly  reduced ;  in  fact,  the  greater  the  current  through  the 
arc  the  greater  the  ionization  and  the  lower  its  resistance.  In 
series  with  the  direct-current  source  of  supply  are  choke  coils  and 
regulating  resistances  which  tend  to  keep  the  supply  current  con- 
stant. Suppose  that  the  arc  is  suddenly  extinguished  and 
deionized.  On  account  of  the  magnetic  energy  in  the  supply  cir- 
cuit the  voltage  across  the  arc  will  rise  very  rapidly,  at  the  same 
time  charging  the  condenser  in  the  oscillatory  circuit  until  a  suffi- 
cient voltage  is  reached  to  strike  the  arc  and  again  ionize  the  path 
of  the  discharge.  The  resistance  of  the  arc  immediately  falls, 

hence  the  condenser  discharges  through 
it,  and  on  account  of  the  inertia  of  the 
discharge,  becomes  charged  again  in  the 
opposite  sense.  It  then  starts  to  dis- 
charge through  the  arc  again,  but  in  the 
opposite  direction  to  the  flow  of  cur- 
rent from  the  source.  Thus,  the  result- 
FIG.  162. — Production  of  high-  ant  current  through  the  arc  is  reduced, 

frequency  currents  by  means  of    and  when  the  discharge   current  of  the 

tne  Poulsen  arc 

condenser  increases  up  to  that  of  the 

supply,  the  resultant  becomes  zero.  At  this  point  the  arc  is  ex- 
tinguished, and,  as  a  result  of  the  features  introduced  by  Poulsen, 
is  rapidly  deionized.  The  supply  current  completes  the  con- 
denser discharge  and  again  charges  up  the  condenser  to  the  point 
where  the  arc  will  strike  again,  and  the  cycle  is  repeated.  It  can 
readily  be  seen  that  while  the  discharge  of  the  condenser  is  de- 
pendent upon  the  natural  period  of  the  oscillatory  circuit,  the 
charging  depends  upon  such  factors  as  arc  length,  constants  of 
the  supply  circuit,  etc.,  so  tliat  the  period  of  the  oscillation  like- 
wise depends  upon  these  latter  factors.  Further,  the  voltage  to 
which  the  condenser  is  charged,  and  hence  the  amplitude  of  the 
oscillation,  depends  upon  the  length  of  the  arc,  rapidity  of  deioniza- 
tion,  etc.,  so  that  the  one  factor  of  arc  length  affects  both  the  fre- 
quency and  intensity  of  the  high-frequency  oscillations. 

In  order  to  obtain  constancy  in  the  oscillations  such  as  is  neces- 
sary for  measuring  purposes  it  is  necessary  that  the  arc  length 
remain  constant.  When  the  arc  is  burning  it  tends  to  eat  into  the 


Radio  Instruments  and  Measurements  223 

electrodes,  and  thereby  increase  its  length,  and  then  move  to 
another  spot  with  a  shorter  gap.  This  results  in  unsteadiness  in 
both  the  frequency  and  amplitude  of  the  oscillations.  In  some 
constructions  the  arc  is  caused  to  revolve  slowly  around  a  cylin- 
drical electrode  by  means  of  a  radial  magnetic  field,  in  others  one 
of  the  electrodes  is  slowly  revolved.  In  either  case  care  must  be 
taken  to  insure  that  the  distance  between  the  electrodes  shall  be 
constant ;  otherwise  slow  changes  in  frequency  and  intensity  will 
result.  In  all  cases  a  transverse  field  used  to  gain  high  power  will 
increase  the  irregularities.  In  general,  the  fluctuations  are  min- 
imized as  the  capacity  in  the  oscillatory  circuit  is  decreased,  the 
wave  length  increased,  and  the  supply  current  increased.  It  is 
practically  impossible  to  attain  reasonable  steadiness  in  the  opera- 
tion at  wave  lengths  much  shorter  than  1000  meters,  though  satis- 
factory operation  is  attainable  at  longer  waves. 

60.  HIGH-FREQUENCY  ALTERNATORS  AND  FREQUENCY  TRANSFORMERS 

The  direct  generation  of  high-frequency  currents  by  means  of 
alternators  is  a  difficult  problem;  in  general,  very  high  speeds  of 
rotation  are  required,  and  the  losses  in  the  machine  from  eddy 
currents,  hysteresis  and  dielectric  absorption  are  likely  to  be  very 
great.  However,  two  types  of  generators  have  been  successfully 
evolved  and  have  been  developed  to  very  high  powers  for  radio 
transmission. 

Inductor  Alternator. — The  first  of  these  is  of  the  inductor  type, 
which  has  been  developed  by  E.  F.  W.  Alexanderson,  of  the  Gen- 
eral Electric  Co.  This  machine  has  stationary  field  and  arma- 
ture windings  and  a  solid  steel  rotor  provided  with  slots  cut  at 
equal  intervals  near  the  circumference  and  filled  with  a  nonmag- 
netic material.  As  the  rotor  revolves  the  magnetic  circuit  of  the 
field  is  closed  alternately  through  the  nonmagnetic  material  filling 
the  slots  and  the  steel  between  the  slots ;  thus  the  magnetic  flux 
due  to  the  field  is  alternately  decreased  and  increased.  This  flux 
threads  the  armature  coils,  setting  up  an  alternating  emf  in  these. 
By  providing  the  rotor  with  300  slots  around  the  circumference 
and  driving  it  at  a  speed  of  20  ooo  revolutions  per  minute  a  fre- 
quency of  100  ooo  cycles  per  second  is  attained.  In  a  later  design 
the  frequency  has  been  increased  to  200  ooo  cycles  per  second. 

It  is  stated  that  there  is  no  difficulty  in  attaining  a  constant 
speed  with  this  machine,  and  hence  it  should  be  of  great  value  for 
measuring  purposes  within  the  range  of  frequencies  covered.  A 

35601°— 18 15 


224  Circular  of  the  Bureau  of  Standards 

further  extremely  valuable  feature  is  that  the  frequency  can  be 
determined  absolutely  from  the  speed. 

Goldschmidt  Alternator. — A  second  type  of  alternator  is  the  so- 
called  reflection  type  due  to  R.  Goldschmidt.  The  rotor  and  stator 
are  each  laminated  and  provided  with  windings.  The  principle 
upon  which  the  operation  of  this  generator  is  based  is  as  follows: 
If  an  alternator  is  excited  with  alternating  current  of  frequency 
Nlt  it  will  generate  current  of  two  frequencies  Nl  +  N2  and 
N!  —  N2  where  N2  is  the  frequency  which  would  be  generated  with 
direct-current  excitation.  If  N1  =  N2  =  N  then  the  frequencies 
would  be  2N  and  o.  If  the  current  of  frequency  2N  is  used  to 
excite  the  field  of  another  similar  generator  running  at  the  same 
speed,  generated  frequencies  of  2N  +  N  and  2N  —  N — that  is,  3-/V 
and  TV — would  result.  Thus,  a  series  of  generators  running  at  a 
moderately  high  speed  could  be  used  for  generating  high-frequency 
currents.  In  the  Goldschmidt  generator  this  frequency  multipli- 
cation is  attained  in  one  machine.  The  stator  is  excited  with  direct 
current  and  current  of  frequency  N  is  generated  in  the  rotor.  Since 
the  induction  of  currents  depends  only  upon  the  relative  motion  of 
rotor  and  stator  we  may  consider  that  the  rotor  field  is  excited 
with  current  of  frequency  N  and  that  the  stator  is  rotating  in  this 
field.  Consequently,  currents  of  frequency  2N  and  o  will  be  gen- 
erated in  the  stator.  The  fields  of  these  currents  in  turn  react 
upon  the  rotor,  producing  in  it  currents  of  frequency  $N  and  N, 
and  in  this  manner  the  frequency  is  successively  stepped  up,  the 
frequencies  in  the  rotor  being  odd  multiples  of  N  and  those  in  the 
stator  even  multiples.  In  order  that  the  flow  of  current  of  these 
frequencies  may  not  be  prevented  by  the  reactance  of  the  circuits, 
the  principle  of  resonance  is  utilized  and  tuned  circuits  are  pro- 
vided for  each  frequency  up  to  that  which  is  to  be  used.  The 
flow  of  current  corresponding  to  the  lower  frequencies  is  sup- 
pressed to  a  great  extent.  For  as  we  have  seen,  starting  with  the 
fundamental  frequency  N,  after  two  ' '  reflections ' '  we  again  have 
an  induced  frequency  N  in  company  with  $N.  It  may  be  shown 
that  these  two  currents  of  frequency  N  will  be  opposite  in  phase  and 
hence  tend  to  neutralize  each  other.  This  is  likewise  true  of  the 
magnetic  fields  so  that  the  losses  due  to  hysteresis  and  eddy  cur- 
rents will  be  caused  mainly  by  the  field  of  the  utilized  frequency 
alone.  While  these  machines  have  been  developed  very  satisfac- 
torily for  radio  transmission  purposes,  it  is  doubtful  whether  they 
could  be  readily  utilized  for  measuring  purposes  in  the  laboratory 


Radio  Instruments  and  Measurements  225 

since  the  multiplicity  of  tuned  circuits  would  render  frequency 
changes  difficult. 

"Static"  Frequency  Transformers. — Several  methods  of  fre- 
quency multiplication  have  been  devised  which  are  based  upon  the 
distortion  of  the  wave  of  magnetic  induction  in  iron  from  that  of 
the  impressed  magnetizing  force.  Since  these  frequency  multi- 
pliers have  no  moving  parts  they  are  called  static  frequency  trans- 
formers. The  principle  is  well  illustrated  in  the  method  of  tripling 
the  frequency,  due  to  Joly.  Fig.  1 63  is  a  typical  curve  showing  the 
variation  of  induction  in  iron  with  the  magnetizing  force.  As  the 
magnetizing  force  is  increased  from  zero  the  resultant  flux  of  induc- 
tion in  the  iron  at  first  increases  rather  slowly,  then  very  rapidly, 
and  then  less  rapidly,  again  becoming  almost  constant  at  a  value 
called  the  saturation  value.  If  the  magnetizing  force  is  alternating 
and  sinusoidal  and  of  such  an  amplitude  that  the  maximum  value 
comes  on  the  steep  part  of  the  induction 
curve  as  at  A ,  Fig.  1 63,  the  resulting  alter- 
nating wave  of  magnetic  induction  will 
be  peaked,  as  in  b,  Fig.  164.  If,  however, 
the  maximum  magnetizing  force  has  a  | 
value  sufficiently  high  to  bring  up  the  in-  J 

duction  to  the  flat  part  of  the  curve  where 

it  is  changing  very  slowly,  as  at  B,  Fig.  Mag^t*^  rmt, 

163,  then  the  resulting  alternating  wave    FlG-  163-—  Variation  of  mag- 

of  induction  will  be  flat  topped,  as  in  c,       ™tic  induciiofn  in  iron  with 

magnetizing  force 

Fig.  1 64.    The  wave  form  b  indicates  that 

there  is  a  strong  harmonic  oscillation  of  three  times  the  fundamental 
frequency  impressed  upon  the  fundamental  oscillation  and  differ- 
ing in  phase  from  it  by  180°.  The  wave  form  c  likewise  indicates 
the  presence  of  a  strong  harmonic  of  three  times  the  fundamental 
frequency  but  which  is  in  phase  with  the  fundamental.  If,  there- 
fore, the  two  waves  b  and  c  can  be  combined  in  such  a  manner  that 
the  fundamental  frequencies  are  180°  out  of  phase  and  hence 
neutralize  each  other,  the  harmonics  of  triple  frequency  will  be  in 
phase  and  will  exist  alone.  This  is  illustrated  in  curve  d  which  is 
obtained  by  subtracting  the  ordinates  of  the  curve  c  from  those  of 
curve  b.  This  method  is  applied  by  means  of  transformers  as 
illustrated  in  Fig.  165.  The  alternator  supplies  current  of  the 
fundamental  frequency  /  to  the  primaries  Pl  and  P2.  Pl  has  few 
turns  and  P2  many  turns,  so  that  the  iron  is  magnetized  more  in- 
tensely in  2  than  in  i.  The  two  secondaries  Sj  and  52  are  so 


226 


Circular  of  the  Bureau  of  Standards 


wound  and  connected  that  the  emf  's  of  the  fundamental  frequency 
neutralize  each  other,  but  the  triple  harmonics  cause  a  current 
flow  in  the  tuned  circuit  of  frequency  3/.  Thus  with  an  initial 
frequency  of  10  ooo  cycles  per  second,  a  frequency  of  30  ooo  can 


FIG.  164. — Method  of  combining  alternating  waves  of 
magnetic  induction  so  as  to  triple  the  frequency 

be  obtained  with  one  transformation.  This  corresponds  to  a 
wave  length  of  10  ooo  meters  and  is  suitable  for  long-distance 
transmission.  Large  powers  may  be  generated. 

For  measuring  purposes  it  would  be  possible  to  step  up  the 
frequency  through  several  stages  obtaining  3,  9,  27,  etc.,  times  the 


FIG.  165. — Use  of  two  transformers  for  producing  fre- 
quency transformations 

fundamental  frequency.  This  might  furnish  a  valuable  method 
of  determining  high  frequencies  in  terms  of  lower  frequencies, 
which  latter  can  be  determined  absolutely  from  the  speed  and 
number  of  poles  of  the  alternator. 


Radio  Instruments  and  Measurements 
61.  BUZZERS 


227 


The  buzzer  is  a  very  convenient  source  of  damped  oscillations 
for  measurement  purposes.  Since,  in  general,  it  furnishes  only 
very  small  power,  it  is  used  in  conjunction  with  very  sensitive 
detecting  instruments.  A  number  of  different  modes  of  connec- 
tion may  be  used  in  generating  oscillations  with  a  buzzer.  That 
shown  in  Fig.  166  has  been  found  to  be  very  satisfactory.  The 
current  from  the  battery  B  flows  through  the  adjustable  resistance 
R,  the  coils  F,  armature  contact 
A,  and  coil  Lt.  When  through 
the  action  of  the  buzzer  the  con- 
tact is  opened,  the  energy  due 
to  the  current  in  the  coil  Lt  is 
transferred  to  the  condenser,  Clt 
giving  it  a  charge.  The  con- 


FiG.   166. — Use  of  buzzer  as  a  source  of 
current  of  definite  frequency 


denser  then  discharges,  causing  a 
train  of  oscillations  in  the  circuit 
Ct  L!,  the  frequency  of  which  depends  upon  the  constants  of  this 
circuit  with  a  small  correction  for  the  capacity  added  by  the  leads, 
etc.,  of  the  buzzer  circuit.  Thus,  each  break  of  the  buzzer  sets 
up  a  train  of  oscillations  in  the  circuit  C\  Lv  The  circuit  L2  C2  is 
a  measuring  circuit  coupled  to  the  driving  circuit  Lt  C\.  The 
current  in  the  measuring  circuit  may  be  indicated  by  a  gal- 
vanometer and  thermoelement  (T)  inserted  directly  in  the 
circuit  or  any  other  sensitive  device. 

Constancy  of  the  high-frequency  cur- 
rent depends  upon  the  steadiness  of  the 
buzzer  action.  This  is  obtained  by  using 
a  good  buzzer  giving  a  note  of  high 
pitch,  such  as  the  Ericsson,  by  adjust- 
ment of  the  buzzer  contacts  and  resist- 
ance R  until  the  buzzer  emits  a  clear 
and  steady  musical  tone,  by  employing 
a  constant  battery,  preferably  a  low- 
voltage  storage  battery  to  insure  steady  direct  current,  and  by 
preventing  sparking  at  the  contact.  This  latter  requirement  is 
attained  by  sending  only  a  moderate  current  through  the  buzzer 
and  by  using  a  fairly  large  fixed  condenser  C3  across  the  buzzer 
field  coils  to  absorb  the  magnetic  energy  stored  therein  which 
otherwise  would  produce  a  high  voltage  and  sparking  at  the  con- 
tact on  break. 


FIG.  167. — Buzzer  circuit  capable 
of  producing  currents  by  shock 
excitation 


228 


Circular  of  the  Bureau  of  Standards 


Another  form  of  buzzer  circuit  which  is  frequently  used  and  is 
capable  of  furnishing  somewhat  larger  currents  is  shown  in  Fig. 
1 67.  In  this  case  the  condenser  C\  is  charged  to  the  voltage  of  the 
battery  when  the  buzzer  contact  is  open  and  discharges  through 
LI  when  the  contact  is  closed.  A  possible  objection  to  this 
circuit  is  the  presence  of  the  buzzer  contact  in  the  oscillatory 
circuit. 

If  Ci  in  the  above  is  a  fixed  condenser  of  several  microfarads 
capacity  and  Ll  a  small  inductance  of  only  one  or  two  turns, 
then  the  oscillations  in  the  circuit  Lt  C\  will  be  very  highly  damped 
and  will  last  for  a  very  short  time,  possibly  only  one  or  two 
oscillations.  Under  these  conditions  an  oscillatory  circuit  coupled 
to  L!  will  be  shocked  into  oscillations  by  what  is  called  impact 
excitation,  the  frequency  and  damping  of  the  oscillations  will 


FIG.  168. — Typical  spark  circuit  for  producing  high-frequency  oscillations 

be  those  natural  to  this  circuit  and  independent  of  the  circuit 
Ll  C\.  On  this  account  this  method  of  impact  excitation  is  very 
useful  in  many  measurements. 

In  place  of  the  very  convenient  buzzer  many  other  forms  of 
circuit  interrupters  may  be  used,  such  as  the  vibrating  wire, 
tuning  fork,  rotating  and  mercury  interrupter. 

62.  THE  SPARK 

Certain  forms  of  spark  gap  are  simple  and  inexpensive  sources 
of  damped  currents  and  so  are  often  used  as  sources  in  high-fre- 
quency measurements.  In  some  kinds  of  measurement  it  is  neces- 
sary or  advantageous  for  the  oscillations  to  have  a  decrement. 

Simple  Spark  Gap. — In  Fig.  168  is  shown  a  typical  circuit  for 
the  generation  of  high-frequency  oscillations  by  means  of  a 
spark  discharge.  The  alternator  supplies  the  low-voltage  wind- 
ing P  of  a  step-up  transformer.  The  high-voltage  side  S  leads 


Radio  Instruments  and  Measurements 


229 


to  the  terminals  of  the  condenser  C,  across  which  is  an  inductance 
L  and  spark  gap  G  in  series.  The  coil  L  is  loosely  coupled  to  the 
measuring  circuit  LmCm  (or  to  the  antenna  in  transmitting). 
During  an  alternation,  as  the  voltage  across  5  increases,  the  con- 
denser C  becomes  charged  up  to  the  point  where  the  voltage  is 
sufficient  to  jump  the  spark  gap.  The  condenser  then  discharges 
through  the  inductance  L  and  the  gap  G.  The  discharge  consists 
of  a  train  of  oscillations  of  a  frequency  approximately  corre- 
sponding to  the  inductance  and  capacity  of  the  circuit.  It  is 
possible  to  adjust  the  voltage  of  the  transformer  and  the  length 
of  the  gap  so  that  the  discharge  takes  place  when  the  voltage  is 
at  a  maximum,  either  positive  or  negative.  In  this  case  one 
spark  and  one  train  of  oscillations  is  obtained  per  alternation 
of  the  supply,  thus  with  a  6o-cycle  generator  the  spark  frequency 


© 


FIG.  169. — Groups  of  oscillations  for  case  of  two  spark  discharges  per  cycle 

will  be  1 20.  By  shortening  the  gap  or  raising  the  voltage  several 
discharges  per  alternation  may  be  obtained.  These  are  called 
partial  discharges  and  occur  somewhat  irregularly.  The  first 
case  is  illustrated  in  Fig.  169.  In  (a)  is  shown  the  transformer 
secondary  voltage  as  it  would  be  if  the  spark  gap  were  absent 
and  in  (6)  the  current  oscillation  in  the  condenser  discharge.  In 
Fig.  170  is  shown  the  effect  of  the  spark  gap  upon  the  damping 
of  the  oscillations  in  the  high-frequency  train.  In  a  circuit  with 
constant  resistance  the  amplitude  would  decrease  exponentially 
as  indicated  by  the  dash  curve  i,  in  the  case  of  a  circuit  with  a 
spark  gap  the  decrease  of  amplitude  tends  to  become  linear,  as 
shown  by  the  dash  line  2.  This  is  due  to  the  increase  in  resistance 
of  the  spark  as  the  amplitude  of  the  current  decreases,  the  effect 
depending  upon  the  material  of  the  electrodes,  etc. 


230 


Circular  of  the  Bureau  of  Standards 


Use  of  Resonance  Transformer. — A  serious  difficulty  in  the 
operation  of  the  spark  circuit  is  caused  by  the  short-circuiting 
of  the  transformer  secondary  by  the  spark.  As  a  result  there  is 
a  heavy  flow  of  current  through  the  gap  causing  the  formation  of 
an  arc  which  reduces  the  amplitude  of  the  oscillations  and  destroys 
the  electrodes.  In  order  to  eliminate  this  difficulty  the  resonance 
transformer  is  used.  The  alternator,  transformer  and  secondary 

condenser  are  adjusted  to  make 
a  system  which  is  in  resonance  for 
the  alternator  frequency.  When 
the  condenser  is  short-circuited 
by  the  spark  the  condition  of 
resonance  is  destroyed,  and  in 
effect  this  is  equivalent  to  the 
sudden  insertion  of  a  reactance 


FlG.  170. — Linear  damping  produced  by 
the  increase  of  spark  resistance  as  the 
amplitude  of  current  decreases 


in  the  transformer  primary.  As 
a  result,  there  is  no  heavy  flow 
of  current  through  the  gap. 
The  theory  of  the  adjustment  of  the  system  of  alternator, 
transformer,  and  secondary  condenser  to  resonance  is  as  follows. 
If  we  have  a  simple  circuit  of  inductance  and  capacity  in  series 
across  the  terminals  of  the  alternator,  as  in  Fig.  171,  the  condi- 
tion for  resonance  for  a  frequency  /  is 


where  Lp  is  the  total  inductance  of  the  circuit  including  the 

alternator.     The    combination    of    transformer    and    secondary 

condenser  can  be  reduced  to  this  simple 

case.     Assuming  that  all  the  induction 

linked  with  the  primary  winding  of  the 

transformer   also    passes    through    the 

secondary  turns — that  is,  that  there  is 

no  magnetic  leakage — and  that  the  ratio 

of  the  number  of  secondary  turns  to 

primary  turns  is  n,  it  may  be  shown 

that  a  capacity  CB  in  the  secondary  is 

equivalent  to  a  capacity  C9  =  n2Cs  in  the  primary.     The  effect  of 

inductance  in  the  secondary  is  decreased  in  the  ratio  of  i :  n2  when 

transferred  back  to  the  primary,  hence  inductance  is  inserted  in 

the  primary  to  tune  to  resonance.     The  total  primary  inductance 


FIG  .171 . — Simp  le  circuit  equiva- 
lent to  Fig.  168 


Radio  Instruments  and  Measurements 


231 


consists  of  that  inserted  plus  the  inductance  of  the  alternator  and 
that  due  to  transformer  leakage.  This  latter  is  small  in  the  case 
of  a  closed-core  transformer.  Experimentally  a  fairly  close  adjust- 
ment to  resonance  may  readily  be  obtained  by  lowering  the  gen- 
erator voltage  until  no  spark  passes  the  gap  and  then  varying 
either  the  primary  inductance  or  secondary  condenser  until  the 
primary  current  or  secondary  voltage  is  a  maximum.  The  primary 
inductance  may  conveniently  consist  of  a  solenoidal  winding  with 
an  iron  core  that  can  be  moved  in  or  out  to  vary  the  inductance  value. 
When  adjusted  to  resonance  the  voltage  across  the  secondary 
condenser  may  rise  to  a  value  much  higher  than  that  correspond- 
ing to  the  voltage  of  the  alternator  and  the  transformer  ratio.  In 
Fig.  172  is  shown  the  way  the  voltage  rises  with  each  alternation 
until  it  is  sufficient  to  jump  the  spark  gap  discharging  the  con- 
denser. The  voltage  then  begins  to  rise  again  until  the  next 
spark  takes  place.  The  alternator  voltage  or  spark  length  can  be 


Spark 


Spark 


Spark 


FIG.  172  — Condenser  voltage  when  the  transformer  system  is  adjusted  to 
resonance  with  the  generator 

adjusted  to  obtain  either  one  spark  per  alternation  or  one  spark 
in  several  alternations,  as  shown  in  the  Fig.  172. 

In  order  to  obtain  constant  high-frequency  current  with  a 
simple  spark  gap  it  is  desirable  to  use  a  low  spark  frequency  in 
order  to  prevent  heating  of  the  gap  which  would  lead  to  arcing. 
Magnesium  electrodes  have  been  found  to  give  the  best  results 
and  to  furnish  oscillations  most  closely  logarithmic  in  damping. 
Zinc  is  also  a  good  material.  The  gap,  the  voltage  and  reso- 
nance conditions  should  be  adjusted  to  give  a  spark  of  moderate 
and  uniform  frequency.  The  alternator  must  run  at  constant 
speed,  otherwise  the  voltage  and  resonance  conditions  will  vary. 
Under  these  conditions  it  is  possible  to  attain  high-frequency 
oscillations  of  a  constancy  which  is  probably  not  excelled  by 
any  other  source  of  damped  oscillations. 


232  Circular  of  the  Bureau  of  Standards 

If  the  resonance  transformer  is  not  utilized,  the  arcing  across 
the  gap  may  be  reduced  by  inserting  resistance  or  inductance 
coils  in  the  primary  of  the  transformer  and  by  employing  an 
air  blast  to  blow  out  the  arc.  Or,  in  place  of  the  simple  gap,  a 
rotary  gap,  as  shown  in  Fig.  173,  may  be  utilized.  Its  character- 
istics are  intermediate  between  those  of  the  simple  and  the 
quenched  spark  gap. 

Quenched  Gap. — It  was  found  by  M.  Wien  that  if  a  series  of 
short  spark  gaps  be  substituted  for  a  single  long  gap  and  a  dis- 
charge passed  through  them,  the  discharge  path 
returns  much  more  quickly  after  discharge  to  its 
initial  condition  of  high  resistance.  This  is  a  result 
of  the  more  rapid  deionization  of  the  gap  and  is 

called  the  quenching  action.     The  quenching  action 
FIG.  iji—Rotat-    .    .  j  .r  ,,  -  r  ,,  .    ., 

ing  spark  gap      1S  lncreased  «  the  surfaces  of  the  gaps  are  of  silver 

or  copper  and  the  gap  is  kept  cool  and  air-tight. 
In  Fig.  1 74  is  shown  a  cross  section  of  a  single  gap  showing  the 
insulating  gasket  between  the  plates  which  renders  the  gap  air- 
tight, the  silver  sparking  surfaces  and  the  flanges  to  provide  a 
large  cooling  surface.  The  insulating  gasket  may  be  of  paper, 
mica  or  rubber,  and  is  about  0.2  mm  thick.  Its  thickness  is 
exaggerated  in  the  figure.  A  number  of  such  gaps  are  stacked 
in  series  and  clamped  together,  and  either  the  leads  to  the  gap 
are  provided  with  clips  so  that  the  number  of  gaps  used  may 
be  varied  or  means  are  provided  for  short-circuiting  as  many 
of  the  gaps  as  desired.  A  plate  of  an  improved  quenched 
gap  designed  at  the  Bureau  of  Standards  is 
shown  in  Fig.  175,  facing  page  323.  The  con- 
struction is  such  as  to  permit  air  circulation  on 
both  sides  of  each  gap.  This  is  accomplished 
by  inverting  alternate  plates.  The  assembled  FlG; 

,  •      T-V«  tion  of  auctioned  gap 

quenched  gap  is  shown  in  Fig.  221,  page  322.  •late 

While  close  coupling  with  the  secondary 
circuit  in  the  case  of  ordinary  spark  gaps  is  to  be  avoided,  since 
it  causes  the  generation  of  two  frequencies  (the  so-called  coup- 
ling waves,  see  p.  48)  of  which  only  one  can  be  utilized,  good  working 
of  the  quenched  gap,  on  the  other  hand,  requires  a  fairly  close  coup- 
ling between  the  primary  and  secondary  circuits.  This  secures  high 
efficiency  and  still  permits  a  single  wave  to  be  obtained.  The  ex- 
planation is  as  follows :  Assume  first  that  the  primary  circuit  contains 
an  ordinary  spark  gap,  the  secondary  (which  may  be  an  antenna)  is 
fairly  closely  coupled  to  the  primary,  and  that  the  two  circuits 


Radio  Instruments  and  Measurements 


233 


when  separated  have  the  same  natural  frequency.  Due  to  the 
coupling,  oscillations  of  two  frequencies,  one  lower  and  one  higher 
than  that  common  to  the  uncoupled  circuits,  will  result  in  both 
circuits  after  the  discharge  takes  place  in  the  primary.  The  com- 
bination of  the  two  frequencies  will  result  in  beats  in  both  cir- 
cuits, the  amplitude  of  the  resultant  oscillation  will  rise  to  a 
maximum  and  fall  to  a  minimum  in  each  circuit,  being  a  maxi- 
mum in  the  primary  when  a  minimum  in  the  secondary,  and  vice 
versa.  As  a  result,  the  total  energy  of  the  oscillations  (excepting 
that  dissipated)  is  transferred  back  and  forth  between  the  two 
circuits.  Although  the  current  in  the  primary  circuit  may  pass 


FIG.  176. — Current  in  (a)  primary  and  (b)  secondary 
when  using  an  ordinary  gap 

through  a  zero  value,  the  rapidity  of  deionization  of  the  ordinary 
spark  gap  is  not  sufficient  to  render  it  nonconducting  in  the 
short  interval  of  time  available  and  the  spark  reignites.  The 
phenomena  are  shown  in  Fig.  176  where  (a)  represents  the  voltage 
oscillations  in  the  primary  and  (6)  the  oscillations  in  the  secondary. 
If,  on  the  other  hand,  a  quenched  gap  is  used  and  the  coupling 
between  the  primary  and  secondary  is  favorable,  it  will  become 
deionized  when  the  primary  oscillations  are  a  minimum  and 
thus  prevent  reignition.  At  this  time  all  of  the  energy  has  been 
transferred  to  the  secondary  and,  since  the  primary  has  become 
inoperative,  this  energy  will  be  dissipated  in  a  train  of  oscilla- 
tions of  which  the  frequency  and  damping  are  determined 


234 


Circular  of  the  Bureau  of  Standards 


entirely  by  the  constants  of  the  secondary  circuit.  The  oscilla- 
tions of  primary  and  secondary  are  shown  in  (a)  and  (6)  of  Fig. 
177.  In  ideal  operation,  the  time  during  which  the  primary 
circuit  is  operative  will  be  extremely  short,  there  will  be  only 
the  one  frequency,  and,  since  the  major  loss  of  power  takes  place 
in  the  high-resistance  primary  circuit,  the  efficiency  will  be  high. 
With  poorer  operation  the  primary  circuit  may  remain  in  opera- 
tion until  the  second  or  third  minimum.  In  this  case  three 


FIG.  177. — Current  in  (a)  primary  and  (6)  secondary  -when  using  aquenched  gap 

waves  may  be  observed,  the  two  coupling  waves  and  the  inter- 
mediate wave  corresponding  to  the  oscillations  of  the  secondary 
by  itself. 

The  connections  for  the  quenched  gap  are  similar  to  those  for 
a  plain  spark,  using  a  resonance  transformer.  Best  operation 
is  generally  obtained  when  the  inductance  in  the  primary  circuit 
is  somewhat  greater  than  that  required  for  resonance.  On  account 
of  the  rapid  quenching  of  the  gap,  the  supply  alternator  may  have 
a  frequency  of  500  cycles  and  adjustments  made  so  as  to  obtain 
one  spark  per  alternation. 


PART  EL—  FORMULAS  AND  DATA 

<& 

CALCULATION  OF  CAPACITY 
63.  CAPACITY  OF  CONDENSERS 

Units.  —  The  capacities  given  by  the  following  formulas  are  in 
micromicrofarads.  This  unit  is  io"12  of  the  farad,  the  farad  being 
defined  as  the  capacity  of  a  condenser  charged  to  a  potential  of  i 
volt  by  i  coulomb  of  electricity.  The  micromicrofarad  and  the 
microfarad  (one-millionth  of  a  farad)  are  the  units  commonly 
used  in  radio  work.  Radio  writers  have  occasionally  used  the 
cgs  electrostatic  unit,  sometimes  called  the  "centimeter."  This 
unit  is  1.1124  micromicrofarads. 

In  the  formulas  here  given  all  lengths  are  expressed  in  centi- 
meters and  all  areas  in  square  centimeters.  The  constants  given 
are  correct31  to  o.i  per  cent. 

PARALLEL  PLATE  CONDENSER 

Let  S  =  surface  area  of  one  plate 
r  =  thickness  of  the  dielectric 

K  =  dielectric  constant  (K  =  i  for  air,  and  for  most  ordinary 
substances  lies  between  i  and  io). 

<^ 
C  =  0.0885^—  micromicrofarads.  (no) 


r 


If,  instead  of  a  single  pair  of  metal  plates,  there  are  N  similar 
plates  with  dielectric  between,  alternate  plates  being  connected 
in  parallel, 

~l)5  (in) 


In  these  formulas  no  allowance  is  made  for  the  curving  of  the 
lines  of  force  at  the  edges  of  the  plates;  the  effect  is  negligible 
when  T  is  very  small  compared  with  5. 

81  The  constants  given  in  the  formulas  are  correct  for  absolute  units.  To  reduce  to  international  units 
the  values  in  absolute  units  should  be  multiplied  by  1.00052.  This  difference  need  not  be  considered  when 
calculations  correct  to  i  part  in  1000  only  are  required. 

235 


236  Circular  of  the  Bureau  of  Standards 

VARIABLE  CONDENSER  WITH  SEMICIRCULAR  PLATES 

Let  N  =  total  number  of  parallel  plates 
r^  =  outside  radius  of  the  plates 
r2  =  inner  radius  of  plates 
T  =  thickness  of  dielectric 
K  =  dielectric  constant 

Then,   for  the  position  of  maximum  capacity   (movable  plates 
between  the  fixed  plates)  , 

-  'W^  (,,2) 


This  formula  does  not  take  into  account  the  effect  of  the  edges 
of  the  plates,  but  as  the  capacity  is  also  affected  by  the  contain- 
ing case  it  will  not  generally  be  worth  while  to  take  the  edge 
effect  into  account. 

Formula  (112)  gives  the  maximum  capacity  between  the  plates 
with  this  form  of  condenser.  As  the  movable  plates  are  rotated 
the  capacity  decreases,  and  ordinarily  the  decrease  in  capacity  is 
proportional  to  the  angle  through  which  the  plates  are  rotated. 

ISOLATED  DISK  OF  NEGLIGIBLE  THICKNESS 

Let  d  =  diameter  of  the  disk 
then  C  =  o.354<i  (113) 


ISOLATED  SPHERE 

Let  d  —  diameter  of  the  sphere 
then  €  =  0.556  d  (114) 

TWO  CONCENTRIC  SPHERES 

Let  rl  =  inner  radius  of  outside  sphere 
r2  =  radius  of  inside  sphere 
K  =  dielectric  constant  of  material  between  the  spheres 

C-I.II2/C—  '-*  (II5) 

v     —  *•  \          <JJ 

'  1         '  2 

TWO  COAXIAL  CYLINDERS 

Let  r1  =  radius  of  outer  cylinder 
r2  =  radius  of  inner  cylinder 

K  =  dielectric  constant  of  material  between  the  cylinders 
/  =  length  of  each  cylinder 

(i  16) 


This  formula  makes  no  allowance  for  the  difference  in  density  of 
the  charge  as  the  ends  of  the  cylinders  are  approached. 


Radio  Instruments  and  Measurements  237 

64.  CAPACITY  OF  WIRES  AND  ANTENNAS. 
SINGLE  LONG  WERE  PARALLEL  TO  THE  GROUND 

For  a  single  wire  of  length  /  and  diameter  d,  suspended  at  a 
height  h  above  the  ground,  the  capacity  is 

o.24i6/ 


Usually  the  diameter  d  may  be  neglected  in  comparison  with 
the  length  /,  and  the  following  equations  are  convenient  for 
numerical  computations. 


•r?      4^  — 

Forest, 


For  -7-Ri, 


c-_ °^L  (II8) 


0.2416  /  (        ^ 

C= ^ (119) 

in  which  the  quantities 


i  + 

>io  • 


and 


may  be  interpolated  from  Table  6,  page  242. 

These  formulas  assume  a  uniform  distribution  of  charge  from 
point  to  point  of  the  wire. 

VERTICAL  WERE 

Formula  (119),  omitting  the  k2  in  the  denominator,  is  sometimes 
used  to  calculate  the  capacity  of  a  vertical  wire.  It  applies 
accurately  only  when  h  is  large  compared  with  Z,  and  gives  very 
rough  values  for  a  vertical  single-wire  antenna,  the  lower  end  of 
which  is  connected  to  apparatus  at  least  several  meters  above  the 
ground. 


238  Circular  of  the  Bureau  of  Standards 

CAPACITY  BETWEEN  TWO  HORIZONTAL  PARALLEL  WIRES  AT  THE   SAME  HEIGHT 

Let  d  =  the  diameter  of  cross  section  of  the  wires 
/  =  length  of  each  wire 
&  =  the  height  of  the  wires  above  the  earth 
D  =  distance  between  centers  of  the  wires. 
The  capacity  is  denned  as  the  quotient  of  the  charge  on  one  wire, 
divided  by  the  difference  in  potential  of  the  two  wires,  when  the 
potential  of  one  wire  is  as  much  positive  as  the  other  is  negative. 

0.1208  1 

(120) 


In  most  cases  d/l  and  D/l  may  be  neglected  in  comparison  with 
unity,  and  we  may  write 

0.1208  /  , 


TWO  PARALLEL  WIRES,  ONE  ABOVE  THE  OTHER 

For  the  case  of  one  wire  placed  vertically  above  the  other,  the 
formula  (121)  may  usually  be  used,  taking  for  the  value  of  h  the 

mean  height  of  the  wrires,  -      —  •     The  potential  of  one  wire  is 
assumed  to  be  as  much  positive  as  the  other  is  negative. 

CAPACITY  OF  TWO  PARALLEL  WIRES  JOINED  TOGETHER 

Let  /  =  the  length  of  each  wire 
D  =  distance  between  centers 
h  =  their  height  above  the  earth 
d  =  diameter  of  cross  section. 

The  wires  are  supposed  to  be  parallel  to  each  other  and  to  lie 
in  a  horizontal  plane.  They  are  joined  together  so  that  they  are 
at  the  same  potential.  The  capacity  is  denned  as  the  quotient 
of  the  sum  of  their  charges  by  the  potential  above  the  earth. 

0-4831  1  ,       N 

(122) 


4*     ,. 
10-    "  '10  J 


/?v 

which,  in  those  cases  where  d*/P  and  I  -r  I   may  be  neglected  in 
comparison  with  unity,  may  be  written  in  the  following  forms: 


Radio  Instruments  and  Measurements 


239 


T?          = 
For      <i, 


0.4831 


logio  ^7 


(123) 


c= 


0.4831  / 


(124) 


The  quantities  kt  and  k2  are  the  same  as  in  (118)  and  (119)  and 
may  be  obtained  from  Table  6,  page  242. 

These  formulas  assume  a  uniform  distribution  of  charge  along 
the  wire.  (See  p.  239.) 

CAPACITY  OF  A  NUMBER  OF  HORIZONTAL  WIRES  IN  PARALLEL 

This  case  is  of  importance  in  the  calculation  of  the  capacity  of 
certain  forms  of  antenna.  The  wires  are  supposed  to  be  joined 
together,  and  thus  all  are  at  the  same  potential.  Their  capacity 
in  parallel  is  then  defined  as  the  quotient  of  the  sum  of  all  their 
charges  by  their  common  potential. 

An  expression  for  this  case  as  accurate  as  the  preceding  formula 
(120)  for  two  wires  would  be  very  complicated.  The  following 
simpler  solution  is  nearly  as  accurate,  and  in  view  of  the  disturbing 
effect  of  trees,  houses,  and  other  like  objects  on  the  capacity  of  an 
antenna,  will  suffice  for  ordinary  purposes  of  design. 
Let  n  =  number  of  wires  in  parallel 

D  =  spacing  of  wires  in  parallel,  measured  between  centers 
d  =  diameter  of  wire 

h  =  height  of  the  wires  above  the  ground 
/  =  length  of  each  wire. 
Then  if  the  potential  coefficients  be  calculated  as  follows: 


/>u  =  4.6o5    ] 
pl2  =  4.605 


or, 


4^ 

'T~    J 

2  h 

rp  -«ij 

=  4.605!  log10^-M 


(I25) 


(126) 


35601°—  1 


-16 


240  Circular  of  the  Bureau  of  Standards 

the  approximate  capacity  of  the  n  wires  in  parallel  will  be 


(127) 


the  quantities  k,  k^  and  k2  being  obtained  from  Tables  6  and  7, 
page  242. 

Example. — To  find  the  capacity  of  an  antenna  of  10  wires  0.16 
inch  in  diameter,  in  parallel,  each  wire  1 10  feet  long,  the  spacing 
between  the  wires  being  2  feet  and  their  height  above  the  ground 
80  feet. 

For  this  case  4&//  =  —  or  7/4/1=0.344   and   Table  6  gives  k 
1 10 

=0.146. 

2XI2XIIO  2/ 

2l/d=~          —2 =  16500,     Iogi0-r     =4.2175 

o.  i  o  a 

"~  =  1.7404 


.-.  pn  =  4.605  [4.218  -0.146]  =  18.75 
pn  =  4.605  [1.740  -0.146]=   7.340 

and  from  formula  (127)  and  Table  7  the  capacity  is,  reducing  the 
length  of  the  wires  to  cm 


=  584  MM/  =0.000584  M/- 

Example.  —  A  second  antenna  of  10  wires,  3/32  inch  diameter, 
155  feet  long,  spaced  2.5  feet  apart,  and  stretched  at  a  distance  of 
64  feet  from  the  earth. 


For  this  case  l/^h  =        =0.606,  k2  =0.249 


2l/d  =  39680,  Iog10^-  =  4- 

l/D  =62,      log10//D  =  1.7924 

P 

'• 


-2.05=6.35 


Radio  Instruments  and  Measurements 


241 


If  the  length  of  the  antenna  had  been  500  feet,  with  the  height 

4h     2  56  dh 

unchanged,  then  V  =  -;-  =0.512,    ^-0.026,    log,0  ^-=4.5154, 

i  5^^ 

logic  ^-1.7093;  by  (125)     pn  =  20.67,  £i2=7-75,  £  =  2.05, 

„     1.112X500X30.5 
C  =  —  -  =  0.002426  fjif. 

6.99 

65.  TABLES  FOR  CAPACITY  CALCULATIONS 
TABLE  5. — For  Converting  Common  Logarithms  Into  Natural  Logarithms 


Common 

Natural 

Common 

Natural 

Common 

Natural 

Common 

Natural 

0 

0.0000 

25.0 

57.  565 

50.0 

115.  129 

75.0 

172.  694 

1.0 

2.  3026 

26.0 

59.  867 

51.0 

117.  432 

76.0 

174.  996 

2.0 

4.  6052 

27.0 

62.  170 

52.0 

119.  734 

77.0 

177.  299 

3.0 

6.9078 

28.0 

64.  472 

53.0 

122.  037 

78.0 

179.  602 

4.0 

9.  2103 

29.0 

66.  775 

54.0 

124.  340 

79.0 

181.  904 

5.0 

11.513 

30.0 

69.  078 

55.0 

126.  642 

80.0 

184.  207 

6.0 

13.  816 

31.0 

71.  380 

56.0 

128.  945 

81.0 

186.  509 

7.0 

16.  118 

32.0 

73.683 

57.0 

131.  247 

82.0 

188.  812 

8.0 

18.  421 

33.0 

75.  985 

58.0 

133.  550 

83.0 

191.  115  ' 

9.0 

20.  723 

34.0 

78.  288 

59.0 

135.  853 

84.0 

193.  417 

10.0 

23.  026 

35.0 

80.590 

60.0 

138.  155 

85.0 

195.  720 

11.0 

25.  328 

36.0 

82.  893 

61.0 

140.  458 

86.0 

198.  022 

12.0 

27.  631 

37.0 

85.  196 

62.0 

142.  760 

87.0 

200.  325 

13.0 

29.  934 

38.0 

87.  498 

63.0 

145.  063 

88.0 

202.  627 

14.0 

32.  236 

39.0 

89.801 

64.0 

147.  365 

89.0 

204.  930 

15.0 

34.539 

40.0 

92.  103 

65.0 

149.668 

90.0 

207.  233 

16.0 

36.841 

41.0 

94.  406 

66.0 

151.  971 

91.0 

209.  535 

17.0 

39.  144 

42.0 

96.  709 

67.0 

154.  273 

92.0 

211.  838 

18.0 

41.  447 

43.0 

99.  Oil 

68.0 

156.  576 

93.0 

214.  140 

19.0 

43.  749 

44.0 

101.  314 

69.0 

158.  878 

94.0 

216.  443 

20.0 

46.  052 

45.0 

103.  616 

70.0 

161.  181 

95.0 

218.  746 

21.0 

48.  354 

46.0 

105.  919 

71.0 

163.  484 

96.0 

221.  048 

22.0 

50.  657 

47.0 

108.  221 

72.0 

165.  786 

97.0 

223.  351 

23.0 

52.  959 

48.0 

110.  524 

73.0 

168.  089 

98.0 

225.  653 

24.0 

55.  262 

49.0 

112.  827 

74.0 

170.  391 

99.0 

227.  956 

100.0 

230.  259 

The  table  is  carried  out  to  a  higher  precision  than  the  formulas,  e.  g.,  2.3026  is  abbre- 
viated to  2.303  in  the  formulas. 

Examples. — To  illustrate  the  use  of  such  a  table,  suppose  we  wish  to  find  the  nat- 
ural logarithm  of  37.48.    The  common  logarithm  of  37.48  is  1.57380. 
If  we  denote  the  number  2.3026  by  M,  then  from  the  table 
i.  5         M=3.  4539 
.  073      M=  .  1681 
.  00080  M—  .  0018 


3.  6238=^37.48 

To  find  the  natural  logarithm  of  0.00748:  The  common  logarithm  is  3-87390,  which 
may  be  written  0.87390—3.     Entering  the  table  we  find 

0.87      M=2.  00325     — 3M=— 6.  9078 
.  0039  M=  .  00898 

sum  2. 0122 

—6.  9078 


—4.  8956  =natural  log  of  0.00748 


242 


Circular  of  the  Bureau  of  Standards 


TABLE  6.— For  Use  in  Connection  with  Formulas  (118),  (119),  (123),  (124),  (125), 

and  (126) 


4h/l 

ki 

l/4h 

k2 

4h/l 

ki 

l/4h 

k2 

0 

0 

0 

0 

0.6 

0.035 

0.6 

0.247 

0.1 

0.001 

0.1 

0.043 

.7 

.045 

.7 

.283 

.2 

.004 

.2 

.086 

.8 

.057 

.8 

.318 

.3 

.009 

.3 

.128 

.9 

.069 

.9 

.351 

.4 

.016 

.4 

.169 

1.0 

.082 

1.0 

.383 

.5 

.025 

.5 

.209 

TABLE  7.— Values  of  k  in  Formulas  (127)  and  (146) 


• 

k 

n 

k 

n 

k 

n 

k 

2 

0 

6 

1.18 

11 

2.22 

16 

2.85 

3 

0.308 

7 

1.43 

12 

2.37 

17 

2.95 

4 

.621 

8 

1.66 

13 

2.51 

18 

3.04 

5 

.906 

9 

1.86 

14 

2.63 

19 

3.14 

10 

2.05 

15 

2.74 

20 

3.24 

CALCULATION  OF  INDUCTANCE 
66.  GENERAL 

In  this  section  are  given  formulas  for  the  calculation  of  self  and 
mutual  inductance  in  the  more  common  circuits  met  with  in  prac- 
tice. The  attempt  is  here  made,  not  to  present  all  the  formulas 
available  for  this  purpose,  but  rather  the  minimum  number  re- 
quired, and  to  attain  an  accuracy  of  about  one  part  in  a  thousand. 
So  far  as  has  seemed  practicable,  tables  have  been  prepared  to 
facilitate  numerical  calculations.  In  some  cases,  to  render  inter- 
polation more  certain,  the  values  in  the  tables  are  carried  out  to 
one  more  significant  figure  than  is  necessary.  In  such  instances, 
after  having  obtained  the  required  quantity  by  interpolation  from 
a  table,  the  superfluous  figure  may  be  dropped.  In  all  the  tables 
the  intervals  for  which  the  desired  quantities  are  tabulated  are 
taken  small  enough  to  render  the  consideration  of  second  differ- 
ences in  interpolation  unnecessary. 

Most  of  the  formulas  given  are  for  low  frequencies,  this  fact  being 
indicated  by  the  subscript  zero,  thus  L0,  M0.  The  high-frequency 
formulas  are  given  where  such  are  known.  Fortunately  it  is 
possible  by  proper  design  to  render  unimportant  the  change  of 
inductance  with  frequency,  except  in  cases  where  extremely  high 
precision  is  required. 

The  usual  unit  of  inductance  used  in  radio  work  is  the  micro- 
henry, which  is  one  millionth  of  the  international  henry.32  The 

w  The  constants  in  the  formulas  for  inductance  given  here  refer  to  absolute  units.  To  reduce  to  inter- 
national units  multiply  by  0.99948.  Since,  however,  an  accuracy  of  the  order  of  only  one  part  in  a  thousand 
is  sought  here,  it  will  not  be  necessary  to  take  this  difference  into  account. 


Radio  Instruments  and  Measurements  243 

henry  is  defined  as  the  inductance  "in  a  circuit  when  the  electro- 
motive force  induced  in  this  circuit  is  one  international  volt,  while 
the  inducing  current  varies  at  the  rate  of  one  ampere  per  second." 
i  henry  =  1000  millihenries  =  io8  microhenries  =  io9  cgs  electro- 
magnetic units. 

^n  the  following  formulas  lengths  and  other  dimensions  are 
expressed  in  centimeters,  unless  otherwise  stipulated,  and  the 
inductance  calculated  will  be  in  microhenries. 

Logarithms  are  given,  either  to  the  natural  base  €  or  to  the 
base  io,  as  indicated.  The  labor  involved  in  the  multiplication 
of  common  logarithms  by  the  factor  2.303  to  reduce  to  the  corre- 
sponding natural  logarithms  will  be  very  materially  reduced  by 
the  employment  of  the  multiplication  table,  Table  5,  page  124, 
which  is  an  abridgement  of  the  table  for  this  purpose  usually  given 
in  collections  of  logarithms. 

All  of  these  formulas  assume  that  there  is  no  iron  in  the  vicinity 
of  the  conductor  or  circuit  of  which  the  inductance  is  to  be  calcu- 
lated. Thus,  the  formulas  here  given  can  not  be  used  to  calculate 
the  inductance  of  electromagnets. 

A  much  more  complete  collection  of  inductance  formulas  with 
numerical  examples  is  given  in  the  Bulletin  of  the  Bureau  of 
Standards,  8,  pages  1-237;  1912>  also  known  as  Scientific  Paper 
No.  169. 

67.  SELF-INDUCTANCE  OF  WIRES  AND  ANTENNAS 

STRAIGHT,  ROUND  WIRE 

If  /  =  length  of  wire 

d  =  diameter  of  cross  section 

ju  =  permeability  of  the  material  of  the  wire 

L0  =  o.oo2/    loge  ~r  — i-f       microhenries  (128) 

=  o.oo2/    2.303  Iog10  -  -  i  +-    microhenries          (129) 
[_  a  4  J 

For  all  except  iron  wires  this  becomes 

L0  =  o.oo2/    2.303  Iog10  ^  -0.75  (130) 

For  wires  whose  length  is  less  than  about  1000  times  the  diameter 

of  the  cross  section  (  -7  <  1000  ),  the  term—,  should  be  added  inside 
\a  /  2/ 

the  brackets.     These  formulas  give  merely  the  self-inductance 


244  Circular  of  the  Bureau  of  Standards 

of  one  conductor.  If  the  return  conductor  is  not  far  away,  the 
mutual  inductances  have  to  be  taken  into  account  (see  formulas 
(134)  and  (136)). 

As   the  frequency   of   the   current   increases,   the   inductance 
diminishes,  and  approaches  the  limiting  value. 


2.303  Iog10  ^  -  i  (131) 


which  holds  for  infinite  frequency. 

The  general  formula  for  the  inductance  at  any  frequency  is 


L=  0.0021    2.303  Iog10  ^  -  i  +M5 


(132) 


where  5  is  a  quantity  given  in  Table  8,  page  282,  as  a  function  of  x 
where 


/3 

VP 


(133) 


/  =  frequency. 

1      p  =  volume  resistivity  of  wire  in  microhm-centimeters 
pc  =  same  for  copper 
/x  =  i  for  all  except  iron  wires. 
For  copper  at  20°  C,  #c=  0.1071  d  -JJ. 

The  value  ac  of  x  for  a  copper  wire  o.  i  cm  in  diameter  at  different 
frequencies  may  be  obtained  from  Table  19,  page  311.  For  a  copper 
wire  d  cm  in  diameter  xc  =  10  d  ac  and  for  a  wire  of  some  other 

material  x  =  10  d  ac  -  I ^  —  • 
V     P 

The  total  change  in  inductance  when  the  frequency  of  the 
current  is  raised  from  zero  to  infinity  is  a  function  of  the  ratio  of 
the  length  of  the  wire  to  the  diameter  of  the  cross  section.  Thus, 
the  decrease  in  inductance  of  a  wire  whose  length  is  25  times  the 
diameter  is  6  per  cent  at  infinite  frequency;  and  for  a  wire  100  ooo 
times  as  long  as  its  diameter,  2  per  cent. 

Example. — For  a  copper  wire  of  length  200  cm  and  diameter 
0.25  cm  at  a  wave  length  of  600  meters,  that  is  /  =  500  ooo,  the  value 
of  x  is  18.93,  and  from  Table  8,  5  =0.037. 

A*  =  i,       =  3200,   Iog10 3200  =  3.51851 


Radio  Instruments  and  Measurements  245 

(From  Table  5) 

loge  3200  =  8.0590 


414 

12 


8.IOI6 

For  zero  frequency 

L0  =  o.4  [8.102  —  i  +0.25]  =  2.  941  microhenry 
For  /  =  500  ooo 

L  =  o.4  [8.  102  -i  +0.037]  =  2.856  microhenry 

a  difference  of  2.9  per  cent  out  of  a  possible  3.4  per  cent. 

For  an  iron  wire  of  the  same  length  and  diameter,  assuming  a 
resistivity  7  times  as  great  as  that  of  copper,  and  a  permeability 

of  loo,  the  value  of  x  is  .*/-  -  times  as  great  as  for  the  copper 
wire,  or  71.5,  and  for  this  value  of  x, 

5=o.oio  (TableS) 
L0  =  o.4  [32.10]  =  12.  84  ph 
L  =0.4  [8.102]  =^.24  nh  at  500  ooo  cycles. 
The  limiting  value  is  Loo  =2.84  ph. 

TWO  PARALLEL,  ROUND  WIRES-RETURN  CIRCUIT 

In  this  case  the  current  is  supposed  to  flow  in  opposite  direc- 
tions in  two  parallel  wires  each  of  length  /  and  diameter  d.  Denot- 
ing by  D  the  distance  from  the  center  of  one  wire  to  the  center 
of  the  other, 


1  2.303  log10-^--j+M5j  (i34) 


L  =  0.004  / 

The  permeability  of  the  wires  being  /x,  and  5  being  obtained  from 
(133)  and  Table  8,  page  282.  For  low  frequency  6=0.25.  This 
formula  neglects  the  inductance  of  the  connecting  wires  between 
the  two  main  wires.  If  these  are  not  of  negligible  length,  their 
inductances  may  be  calculated  by  (132)  and  added  to  the  result 
obtained  by  (134),  or  else  the  whole  circuit  may  be  treated  by 
the  formula  (138)  for  the  rectangle  below. 


246  Circular  of  the  Bureau  of  Standards 

STRAIGHT  RECTANGULAR  BAR 

Let  /  =  length  of  bar. 
b,  c  =  sides  of  the  rectangular  section. 

L0  =0.002  /  £2.303  Iog10  ^  +0.5  +0.2235    (-~^  (I35) 


The  last  term  may  be  neglected  for  values  of  /  greater  than 
about  50  times  (b  +  c)  . 

The  permeability  of  the  wire  is  here  assumed  as  unity. 

RETURN  CIRCUIT  OF  RECTANGULAR  WIRES 

If  the  wires  are  supposed  to  be  of  the  same  cross  section,  6  by 
c,  and  length  /,  and  of  permeability  unity,  and  the  distance  be- 
tween their  centers  is  D, 


L0  =  0.004  ^2.303  loSio  b^~c  +  l~l+  a2235  ^y^    I    (136) 


FIG.  178. — The  two  conductors 
of  a  return  circuit  of  rectan- 
gular wires 

For  wires  of  different  sizes,  the  inductance  is  given  by  L0=Lt  + 
L2  —  2M  in  which  the  inductances  Lt  and  L2  of  the  individual  wires 
are  to  be  calculated  by  (135),  and  their  mutual  inductance  M  by 
(174)  below. 

SQUARE  OF  ROUND  WIRE 

If  a  is  the  length  of  one  side  of  the  square  and  the  wire  is  of 
circular  cross  section  of  diameter  d,  the  permeability  of  the  wire 
being  M, 

L=o.oo8  a  I  2.303  log10-^  +  — -0.774+M5J  (137) 

in  which  8  may  be  obtained  from  Table  8  as  a  function  of  the 
argument  x  given  in  formula  (133).  The  value  of  d  for  low  fre- 
quency is  0.25,  and  for  infinite  frequency  is  o. 


Radio  Instruments  and  Measurements 


247 


RECTANGLE  OF  ROUND  WIRE 

Let  the  sides  of  the  rectangle  be  a  and  alf  the  diagonal 
g  =  Vo2  +  a?  and  d  =  diameter  of  the  cross  section  of  the  wire. 
Then  the  inductance  at  any  frequency  is 


L=  0.0092 1    (a  +  ajlogj 
L 
+  0.004  [M^  (a  +  a^+2  (g  +  d/2)—2  (a  +  aj]  (138) 

The  quantity  6  is  obtained  by  use  of  (133)  and  Table  8.     Its 
value  for  zero  frequency  is  0.25,  and  is  o  for  infinite  frequency. 

RECTANGLE  OF  RECTANGULAR-SECTION  WIRE 


FIG.  179.  —  Rectangle  of  rectan- 
gular -wire 

Assuming  the  dimensions  of  the  section  of  the  wire  to  be  b 
and  c,  and  the  sides  of  the  rectangle  a  and  alt  then  for  nonmag- 
netic material  the  inductance  at  low  frequency  is 


=  0.0092  1 


where     ^- 


I 


Iog1 


+  0.004 


[,- 


-a  Iog10  (a  +  g)  -al  Iog1 


-  +  0.447  (b+c) 


(139) 


INDUCTANCE  OF  GROUNDED  HORIZONTAL  WIRE 


If  we  have  a  wire  placed  horizontally  with  the  earth,  which 
acts  as  the  return  for  the  current,  the  self-inductance  of  the  wire 
is  given  by  the  following  formula,  in  which 

I  =  length  of  the  wire 

h  =  height  above  ground 

d  =  diameter  of  the  wire 

ju  =  permeability  of  the  wire 

5  =  constant  given  in  Table  8,  to  take  account  of  the  effect  of 
frequency  (see  p.  282). 


[ 


4k 


d~\ 
- 


/       x 


248  Circular  of  the  Bureau  of  Standards 

which,  neglecting  j ,  as  may  be  done  in  all  practical  cases,  may  be 
written  in  the  following  forms  convenient  for  calculation: 
For  y^i, 

A.H 

L  =  o.oo2  l\  2.3026  log10-T-  — P  +  /IO  (141) 

and  for  -7^1, 
2h 

L  =  0.002  /    2.3026  Iog10  ~-Q+tJ,8  (142) 

the  values  of  P  and  Q  being  obtained  by  interpolation  from 
Table  9. 

Mutual  Inductance  of  Two  Parallel  Grounded  Wires. — The  two 
wires  are  assumed  to  be  stretched  horizontally,  with  both  ends 
grounded,  the  earth  forming  the  return  circuit. 
Let  I  =  length  of  each  wire 
d  =  diameter  of  wire 

D  =  distance  between  centers  of  the  wires 
h  =  height  above  the  earth 
Then 


£    _r     ^h*+D2 

=  o.oo46o5  /  [loglo  *  L_  _  +  log 


+  0.002  /  [~y//2  +  D2  +  4/fc2  —  V^2  +  D2  +D  —  -\/D2  +  4/£2]          (143) 

which,  if  we  neglect  -^  and  (  -r  )2  may  be  expressed  in  the  follow- 
ing forms : 
For  y<i, 

[j  rf*l 

2.3026  Iog10  ^  -P+^J  (144) 


M 


and  for  — f<i, 
2h 


M  =  0.002  /    2.3026  Iog10  7)-<2+7  (J45) 

the  values  of  the  quantities  P  and  Q  being  obtained  by  interpo- 
lation from  Table  9. 


Radio  Instruments  and  Measurements  249 

INDUCTANCE  OF  GROUNDED  WIRES  IN  PARALLEL 

The  expressions  for  the  inductance  of  n  grounded  wires  in  par- 
allel involve  the  inductances  of  the  single  wires  and  the  mutual 
inductances  between  the  wires.  Even  in  the  case  that  the  wires  are 
all  alike  and  evenly  spaced,  these  expressions  are  very  complicated. 

The  following  approximate  equation,  which  neglects  the  resist- 
ances of  wires,  is  capable  of  giving  results  accurate  to  perhaps  i 
per  cent,  for  n  wires  of  the  same  diameter  evenly  spaced. 

Calculate  by  equations  (141),  (142),  (144),  or  (145)  the  induc- 
tance L!  per  unit  length  of  a  single  wire  and  the  mutual  induc- 
tance Ml  per  unit  length  of  any  two  adjacent  wires  using,  of 
course,  the  actual  length  in  the  calculation  of  the  ratios 

2/Z,     2l  - 

-j->  -j>  etc.     Then 

/     a 

T      .TLi  +  Cn-i)  Mt  ,"] 

L  =  l\      —  -  -  -  -  -o.ooikl  (146) 

in  which  n  is  the  number  of  wires  in  parallel  and  k  is  a  function 
of  n  tabulated  in  Table  7,  page  242. 

Example.  —  An  antenna  of  10  wires  in  parallel,  each  wire  155 
feet  long  and  -fa  inch  in  diameter,  spaced  2.5  feet  apart,  and  sus- 
pended at  a  height  of  64  feet  above  the  earth.  Find  the  inductance 
at  100  ooo  cycles  per  second. 

We  have  here  -r-=—;  —  =  0.826,  and  using  this  as  argument  in 

^J  \j 

Table  9,  P=  0.6671. 

From  (133)  x  =  8.07,  and  thence  from  Table  8,  6=0.087. 


i2X       =32768,  Iog10     =  4. 
128 


logi<fp  =1.709 


Then,  from  formulas  (141)  and  (144) 


L1=o.oo2[4.5i5  X  2.  3026  -0.667  +0.087] 
==0.01963  nh  per  cm 

M1=o.oo2[i.  709x2.  3026  —  0.667+0.016] 
=  0.006568  p.h  per  cm. 


250  Circular  of  the  Bureau  of  Standards 

From  Table  7  we  find  for  n  =  10,  k  —  2.05,  so  that  the  inductance 
as  calculated  by  (146)  is 


,  [""0.01963+9(0.006568) 

£  =  155X30.5  ~^~    -^-^ -0.00205 1 

-=4727  [0.00582]  =27.4  ph. 

CIRCULAR  RING  OF  CIRCULAR  SECTION 

If  a  =mean  radius  of  ring 

d  =  diameter  of  wire,  the  inductance  at  any  frequency  is, 

except  for  values  of  —  >  0.2 , 


L  =0.01257  a  {2.303  Iog10-j — 2+/x5[  (147) 

I  »  J 

in  which  8  will  be  obtained  from  (133)  and  Table  8,  page  282. 
Its  value  for  zero  frequency  is  0.25. 

TUBE  BENT  INTO  A  CIRCLE 

Let  the  inner  and  outer  diameters  of  the  annular  cross  section 
of  the  tube  be  dl  and  d2,  respectively,  and  the  mean  radius  of  the 

circle  a,  then  neglecting  -^  and  -~ 

i6a  d,2 

L0  =  0.01257  a  I  2.303  Iog10  "^~~I-75~2M2_fj2) 


For  infinite  frequency  this  becomes 

LOO  =0.012570!  2.303  log10^-2J  (i49) 

68.  SELF-INDUCTANCE  OF  COILS 

CIRCULAR  COIL  OF  CIRCULAR  CROSS  SECTION 

For  a  coil  of  n  fine  wires  wound  with  the  mean  radius  of  the 
turns  equal  to  a,  the  area  of  cross  section  of  the  winding  being  a 
circle  of  diameter  d, 


i6a 

an2    2.303  Iog10—  r-i-75 
I  a  ) 


Radio  Instruments  and  Measurements  251 

This  neglects  the  space  occupied  by  the  insulation  between  the 


wires. 


TORUS  WITH  SINGLE-LAYER  WINDING 


A  torus  is  a  ring  of  circular  cross  section  (doughnut  shape) . 
Let  R  —  distance  from  axis  to  center  of  cross  section  of  the  winding 
o  =  radius  of  the  turns  of  the  winding 
n  =  number  of  turns  of  the  winding 

(151) 


FIG.  180. — Torus  of  single  layer 
•winding 

TOROIDAL  COIL  OF  RECTANGULAR  CROSS  SECTION  WITH  SINGLE-LAYER  WINDING 

A  coil  of  this  shape  might  also  be  called  a  circular  solenoid  of 
rectangular  section. 

Let  rx  =  inner  radius  of  toroid  (distance  from  the  axis  to  inside 

of  winding) 
r2  =  outer  radius  of  toroid  (distance  from  axis  to  outside  of 

winding) 
h  =  axial  depth  of  toroid. 


Then  L0  =  0.004606  n2  h  logt 


(152) 


FIG.  181. — Toroidal  coil  of  rec- 
tangular section  with  single 
layer  winding 

The  value  so  computed  is  strictly  correct  only  for  an  infinitely 
thin  winding.  For  a  winding  of  actual  wires  a  correction  may  be 
calculated  as  shown  in  Bulletin,  Bureau  of  Standards,  8,  page 
125;  1912.  The  correction  is,  however,  very  small. 


252  Circular  of  the  Bureau  of  Standards 

SINGLE-LAYER  COIL  OR  SOLENOID 

An  approximate  value  is  given  by 

L^_ 0.03948  o«n'g  (I53) 

where  n  =  number  of  turns  of  the  winding,  a  =  radius  of  the  coil, 
measured  from  the  axis  to  the  center  of  any  wire,  b  =  length  of 
coil  =  n  times  the  distance  between  centers  of  turns,  and  K  is  a 

2a 

function  of  -r-  and  is  given  in  Table  10,  page  283,  which  was  calcu- 
lated by  Nagaoka.  (See  Bulletin,  Bureau  of  Standards,  8,  p. 
224,  1912.)  For  a  coil  very  long  in  comparison  with  its  diameter, 
JC-x, 

Formula  (153)  takes  no  account  of  the  shape  or  size  of  the  cross 
section  of  the  wire.  Formulas  are  given  below  for  more  accurate 
calculation  of  the  low-frequency  inductance.  The  inductance 
at  high  frequency  can  not  generally  be  calculated  with  great 
accuracy.  Formulas  which  take  account  of  the  skin  effect,  or 
change  of  current  distribution  with  frequency,  have  been  devel- 
oped. The  change  is  very  small  when  the  coil  is  wound  with 
suitably  stranded  wire.  The  inductance  at  high  frequencies 
depends,  however,  also  on  the  capacity  of  the  coil,  which  is  gen- 
erally not  calculable.  If  the  capacity  is  known,  from  measure- 
ments or  otherwise,  its  effect  upon  the  inductance  can  be  cal- 
culated by 

La  =  L[i+co2CL(io)-'«]  (154) 

where  La  is  the  apparent  or  observed  value  of  the  inductance,  C 
is  in  micromicrofarads,  and  L  in  microhenries.  The  inductance 
of  a  coil  is  decreased  by  skin  effect,  and  is  increased  by  capacity. 
The  changes  due  to  these  two  effects  sometimes  neutralize  each 
other,  and  in  general,  formula  (153)  gives  about  as  good  a  value 
of  the  high-frequency  inductance  as  can  be  obtained. 

Round  Wire. — The  low-frequency  inductance  of  a  coil  wound 
with  round  wire  can  be  calculated  to  much  higher  precision  than 
that  of  formula  (153)  by  the  use  of  correction  terms.  Formula 
(153)  gives  strictly,  the  inductance  of  the  equivalent  current 
sheet,  which  is  a  winding  in  which  the  wire  is  replaced  by  an  ex- 
tremely thin  tape,  the  center  of  each  turn  of  tape  being  situated 
at  the  center  of  a  turn  of  wire,  the  edges  of  adjacent  tapes  being 
separated  by  an  infinitely  thin  insulation.  The  inductance  of  the 
actual  coil  is  obtained  from  the  current-sheet  inductance  as 
follows : 


Radio  Instruments  and  Measurements  253 

Putting  L8  =  inductance  of  equivalent  cylindrical  current  sheet, 

obtained  from  (153) 

Lo  =  inductance  of  the  coil  at  low  frequencies 
n  =  number  of  turns 

a  =  radius  of  coil  measured  out  to  the  center  of  the  wire 
D  =  pitch  of  winding  =  distance  from  center  of  one  wire 
to  the  center  of  the  next  measured  along  the  axis 
b  =  length  of  equivalent  current  sheet  =  nD 
d  =  diameter  of  the  bare  wire 

Then         L0  =  L8  — 0.01257  na  (A  +  B)  microhenry  (155) 

in  which  A  is  constant,  which  takes  into  account  the  difference  in 
self-inductance  of  a  turn  of  the  wire  from  that  of  a  turn  of  the 
current  sheet,  and  B  depends  on  the  difference  in  mutual  induc- 
tance of  the  turns  of  the  coil  from  that  of  the  turns  of  the  current 
sheet.  The  quantities  A  and  B  may  be  interpolated  from  Tables 
ii  and  12,  page  284,  which  are  taken  from  Tables  7  and  8  of  Bul- 
letin, Bureau  of  Standards,  8,  pages  197-199;  1912. 

Example. — A  coil  of  400  turns  of  round  wire  of  bare  diameter 
0.05  cm,  wound  with  a  pitch  of  10  turns  per  cm,  on  a  form  of  such 
a  diameter  that  the  mean  radius  out  to  the  center  of  the  wire  is 
10  cm. 

^o,   ^  =  400,   D  =  o.i,—  =0.5 

2/r 

The  value  of  K  corresponding  to  -r-  =  o.  5  is  0.8 1 8 1  (Table  10) . 

IOO 

Ls  =0.03948  (4Oo)2  -  -  0.8181  =0.03948X400  000x0.8181 
4° 

=  12  919  microhenries 
=0.012919  henry 

log  0.03948  =  2.59638 

log  400  ooo  =  5.60206 

log    0.8181=1.91281 


4.11125 
Entering  Tables  1 1  and  1 2  with  -^  =  0.5,  n  =  400,  we  find 

A  =  —0.136 
B=     0.335 


A+B=     0.199 
The  correction  in  (155)  is,  accordingly 

0.01257  (400)  10  (0.199)  =  9-99  microhenries. 


254  Circular  of  the  Bureau  of  Standards 

The  total  inductance  is  12  919  —  10  =  12  909  microhenries. 

Example.  —  A  coil  of  79  turns  of  wire  of  about  0.8  mm  bare 
diameter.  The  mean  diameter  is  about  22.3  cm  and,  for  deter- 
mining the  pitch,  it  was  found  that  the  distance  from  the  first 
to  the  79th  wire  was  9.0  cm. 

We  have,  then, 


b  =nD  =  79X0.115  =9.12 


2a__  d  _  0.08 

=   M45'       = 


The  value  of  K  is  given  by  Table  10  as  0.4772,  so  that 

(n  is)2 
Lg  =0.03948  (79)2  •-  —  '—^-  0.4772  =  1602.8  microhenries 

log  0.03948  =  2.  59638  ^  d  ^  , 

For  n  =  79,  7^=0.7,  Tables  n 
2  log          79=3.79526  D 

2  log      11.15=2.09454  and  12  give 

log    0.4772=1.67870  A  =o.  200 

5=0.326 

4.  16488 

log        9.12=0.95999  (A  +B)  =0.526 


3- 20489 

The  correction  is  0.01257  X  79  X  11.15  X  0.526  =  5. 8  microhen- 
ries, and  the  total  is  1597.0  microhenries.  The  measured  in- 
ductance of  this  coil  is  1595.5. 

COIL  WOUND  WITH  WIRE  OR  STRIP  OF  RECTANGULAR  CROSS  SECTION 

Approximate  values  may  be  obtained  for  a  coil  wound  with 
rectangular-section  wire  or  strip  by  using  the  simple  formula 
(153),  as  already  explained.  More  precise  values  for  the  low- 
frequency  inductance  could  be  calculated  in  the  same  manner 
as  for  round  wire  above,  using  different  values  for  A  and  B.  It 
is  simpler,  however,  to  use  formula  (156)  below,  which  applies 
to  the  single-layer  coil,  if  the  symbols  are  given  the  following 
meaning:  a  =  radius  measured  from  the  axis  out  to  the  center  of 
the  cross  section  of  the  wire;  6=  the  pitch  of  the  winding  D, 
multiplied  by  the  number  of  turns  n;  c=w  =  the  radial  dimen- 
sion of  the  wire;  Z=the  axial  thickness  of  the  wire.  The  cor- 
rection for  the  cross  section  of  the  wire  is  obtained  by  using 

iv         t 
formulas  (161)  and  (162),  using  v  =  ^>  T=r\' 


Radio  Instruments  and  Measurements  255 


Example.  —  A  solenoid  of  30  turns  is  wound  with  ribbon  J4 
by  iV  inch  thick,  with  a  winding  pitch  of  X  inch  to  form  a  sole- 
noid of  mean  diameter  10  inches. 

Here  0  =  5X2.54  =  12.70  cm,  w=c  =  -  (2.54)  =0.635  cm 

4 

6  =  30X^(2.  54)  =19.05  cm,  c/b  =  ^,D=  0.635 


for  the  equivalent  coil.     Solving  this  by  Rosa's  formula  (156), 

using  ^  =  ->#  =0.6230  (Table  10),  -  =  30,  ^,=0.693,  #8=0.3218, 
03  ^ 

we  find  Lu  =  182.55  fj.h.     The  value  obtained  by  Stefan's  formula 
(157)  is  very  slightly  in  error,  being  182.5. 

w  i 

To  obtain  the  correction,  we  have  v  =  j=:  =  i,  r  =  ->  and  therefore 

u  4 

=  0.470 


1.25 


'.-D 


2Q  28  27  26 

B.=  -2 1    —  0.060  H 0.018  +-—  0.008  H 0.005 

1  30  30  30  30 

21  1 

+    .    .    .+ — o.ooi    = -0.188 
30 


so  that  the  correction  is  (0.01257)  30  (12.70)   (0.285)  =1.36  /*&» 
and  the  total  inductance  is  183.9. 

INDUCTANCE  OF  POLYGONAL  COILS 

Such  coils,  instead  of  being  wound  on  a  cylindrical  form,  are 
wrapped  around  a  frame  such  that  each  turn  of  wire  incloses  an 
area  bounded  by  a  polygon. 

No  formula  has  been  developed  to  fit  this  case,  but  it  is  found 
that  the  inductance  of  such  a  coil  (when  the  number  of  sides  of 
the  polygon  is  fairly  large)  may  be  calculated,  within  i  per  cent, 
by  assuming  that  the  coil  is  equivalent  to  a  helix,  whose  mean 
radius  is  equal  to  the  mean  of  the  radii  of  the  circumscribed  and 
inscribed  circles  of  the  polygon.  That  is,  if  r  =  the  radius  of 
the  circumscribed  circle,  Fig.  182  (which  can  be  measured  without 
difficulty  for  a  polygon  for  which  the  number  of  sides  N  is  an 

even  number),  then  the  modified  radius  a0 —r  cos2  -^  is  to  be  used 
for  a  in  the  formulas  (153)  and  (155)  of  the  preceding  section. 

35601°— 18 17 


256 


Circular  of  the  Bureau  of  Standards 


Examples. — The  following  table  gives  the  results  obtained  by 
this  method  for  some  1 2-sided  polygonal  coils,  the  measured 
inductance  being  given  for  comparison.  For  N  =  i2,  a0  = 


Coil 

r 

Oo 

n 

D 

b 

L0 
calculated 

M* 

Lo 
measured 
P* 

A 

6.35 

6.24 

23 

0.32 

7.3 

63.0 

61.7 

B 

a  25 

8.10 

28 

.32 

9.0 

124.7 

126.3 

C 

11.43 

11.22 

52 

.212 

11.0 

638.0 

630.5 

D 

11.43 

11.22 

34 

.388 

10.8 

274.9 

274.6 

E 

13.97 

13.73 

64 

.211 

13.1 

1119.  5 

1115.5 

F 

19.05 

18.71 

117 

.158 

18.5 

5389 

• 

5387 

MULTIPLE-LAYER  COILS 


Different  formulas  are  used  for  long  than  for  short  coils. .  Far 
long  coils  of  few  layers,  sometimes  called  multiple-layer  solenoids, 
the  inductance  is  given,  approximately,  by 


,  . 

(0.693+5.) 


(156) 


FIG.  182. — Polygonal  coil 

where  L8  =  inductance,  calculated  by  (153),  letting 
n  =  number  of  turns  of  the  winding 
a  =  radius  of  coil  measured  from  the  axis  to  the  center  of 

cross  section  of  the  winding 
b  =  length   of   coil  =  distance   between   centers   of   turns, 

times  number  of  turns  in  one  layer 
c  =  radial  depth  of  winding  =  distance  between  centers  of 

two  adjacent  layers  times  number  of  layers 
B6  —  correction  given  in  Table  13,  page  284,  in  terms  of  the 

ratio  - 
c 


Radio  Instruments  and  Measurements 


257 


Values  obtained  by  this  formula  are  less  accurate  as  the  ratio 
c/a  is  greater,  and  may  be  a  few  parts  in  1000  in  error  for  values  of 

this  ratio  as  great  as  0.25,  and  -  as  great  as  5.     For  accurate  results 

a  correction  needs  to  be  applied  to  Lu(see  (159)  below). 

The  solution  of  the  problem  for  short  coils  is  based  on  that  for 
the  ideal  case  of  a  circular  coil  of  rectangular  cross  section.  Such 
a  coil  would  be  realized  by  a  winding  of  wire  of  rectangular  cross 


'Axis 


FIG.  183. — Multiple-layer  coil  with 
winding  of  rectangular  cross 
section 

section,  arranged  in  several  layers,  with  an  insulating  space  of 
negligible  thickness  between  adjacent  wires. 

Let  a  =  the  mean  radius  of  the  winding,  measured   from  the 

axis  to  the  center  of  the  cross  section 
6=  the  axial  dimension  of  the  cross  section 
c  =  the  radial  dimension  of  the  cross  section 
d  =  V&2  +  c2  =  the  diagonal  of  the  cross  section 
n  =  number  of  turns  of  rectangular  wire. 

Then,  if  the  dimensions  b  and  c  are  small  in  comparison  with  a, 
the  inductance  is  very  accurately  given  by  Stefan's  formula,  which, 
for  b  >  c,  takes  the  form 


Lu  =  0.01257  an2 


b2 


where  yl  and  y2  are  constants  given  in  Table  14,  page  285. 


\ 


258  Circular  of  the  Bureau  of  Standards 

For  disk  or  pancake  coils,  b<c,  and  the  formula  becomes 

.    .      .     -      .     ~     \H      '  8a  c2 

Lu  =  o.oi257att2 

=  0.01257  aw2 


in  which  yt  and  y3  are  given  in  Table  14,  page  285. 

The  constant  yt  is  the  same  function  of  both  b/c  and  c/b,  so  that 
its  argument,  in  any  given  case,  is  the  ratio  of  the  smaller  dimen- 
sion to  the  larger;  y2  and  y3  are  functions  of  c/b  and  b/c,  respect- 
ively, the  arguments  being  not  greater  than  unity  in  either  case. 

The  error  due  to  the  neglect  of  higher  order  terms  in  -  and  -  in 

€*•  d 

formulas  (157)  and  (158)  becomes  more  important  the  greater  the 
diagonal  of  the  cross  section  is,  in  comparison  with  the  mean 
radius,  but  even  in  the  most  unfavorable  case,  c/b  small,  the  inac- 
curacy with  values  of  the  diagonal  as  great  as  the  mean  radius 
does  not  exceed  one-tenth  of  i  per  cent.  The  accuracy  is  greater 
with  disk  coils  than  with  long  coils,  and  best  of  all  when  tne 
cross  section  is  square. 

For  long  coils  (those  in  which  the  length  b  is  greater  than  the 
mean  radius  a) ,  the  error  of  formula  (157)  becomes  rapidly  greater. 
In  cases  where  both  dimensions  of  the  cross  section  are  large,  in 
comparison  with  the  mean  radius,  no  formulas  well  adapted  to 
numerical  computations  are  available,  but  this  is  not  to  be  regarded 
as  a  case  of  practical  importance  in  radio  engineering. 

COIL  OF  ROUND  WIRE  WOUND  IN  A  CHANNEL  OF  RECTANGULAR  CROSS  SECTION 

If  we  suppose  that  the  distance  between  the  centers  of  adjacent 
wires  in  the  same  layer  is  Dlt  and  that  the  distance  between  the 
centers  of  wires  in  adjacent  layers  is  D2,  then  the  dimensions  of  the 
cross  section  of  the  equivalent  coil  with  uniform  distribution  of 
the  current  over  the  cross  section  will  be  given  by  b  =  wxD, ,  c  =  n2D2, 
where  nt  and  n2  are,  respectively,  the  number  of  turns  per  layer, 
and  the  number  of  layers. 

The  inductance  of  the  equivalent  coil  calculated  by  formulas 
(156),  (157),  or  (158),  using  these  dimensions  and  the  same  mean 
radius  as  the  actual  coil,  is  a  very  close  approximation  to  the 
value  for  the  actual  coil,  unless  the  percentage  of  the  cross  section 
occupied  by  insulating  space  is  large. 


Radio  Instruments  and  Measurements  259 

When  such  is  the  case,  the  correction  to  the  inductance,  given 
in  the  following  formula,  may  be  added: 


AL=  0.01257  an    2.30  Iog10  ^  +  o.i38+E  (159) 

in  which  D  =  distance  between  centers  of  adjacent  wires 
d  =  diameter  of  the  bare  wire 

E  =  a  term  depending  on  the  number  of  turns  and  their 

•  arrangement  in  the  cross  section.     Its  value  may  with  sufficient 

accuracy  be  taken  as  equal  to  0.017.     The  correction  in  (159) 

should,  in  any  case,  be  roughly  calculated,  to  see  if  it  need  be 

taken  into  account. 

Example.  —  Suppose  a  coil  of  winding  channel  6=c  =  i-5  cm, 
wound  with  15  layers  of  wire,  with  15  turns  per  layer,  the  mean 
radius  of  the  winding  being  5  cm.     Diameter  of  bare  wire  =  0.08  cm. 
In  this  case 

d2     A.  ^ 
=  4.5,—  =  ^=0.18,  bfc  =  i,  ^=0.8483,^3  =  0. 

>'log.  - 


—  fc\  r~»T  o  c  "7^  t 

;5)(225y2pI  + 

3(o.3)2  +  (o.3)2 

u          ^U.Ul  Z$  1  )  \ 

96 

-0.8483+- 

^o.8i6J 

log  8       =0.90309 

2.76310 

1.00375  lo 

\  log  0.18=7.62764 

17269 
104 

8a 
•~d  =      2>  9478 

-y,=    •   .8483 
104 

8a  8a  0.09    0  , 

-j     =1.27545     2. 93683=  log.  -^-  -^0.8 1 6 

2.  104 


Iog102.i04     =o-323°5 

log10225  =4.70436 

Iog10  o.oi  257  =  2.09934  Lu  =  6694  microhenries. 

Iog105  =0.69897 


3.82572 
The  correction  for  insulation  is  found  from  (159),  as  follows: 

f-IS'-f'  log.of  =°.0969I,  log.|-0.223 

0.138 

E  =  o.  017 


0.378 
correction  =  (0.0125 7)  (5)  (225)  0.378  =3.34^ 


260  Circular  of  the  Bureau  of  Standards 

The  total  inductance  is  6697  microhenries  =  6. 69 7  millihenries. 
The  correction  could,  in  this  case,  have  been  safely  neglected. 
Example. — A  coil  of  10  layers  of  100  turns  per  layer,  mean 
radius  =  10  cm,  the  wires  being  spaced  o.i  cm  apart. 
For  this  case  n  =  1000,  a  =  10,  b  =  10,  c  =  i . 

2a 
Using    formula    (156)    with    -r-  =  2,      K  =  0.5255,      b(c  =  io 


L9  =  (0.03948)  l~  *   IO  0.5255  =207  400  microhenries. 
10 

For  the  correction,  Table  13  gives  for  -  =  io 

C 

0.693 

Ba =0.279 


0-973 
so   that    the  correction  =  (0.01257)  io6  —  0.973  =  12  200    and    the 

inductance  is 

Lu  =  207  400  —  1  2  200  =  195  200  microhenries 
=  195.2  millihenries. 

The  formula  (157)  gives  a  value  about  one  part  in  900  higher 
than  this. 

INDUCTANCE  OF  A  FLAT  SPIRAL 

Such  a  spiral  may  be  wound  of  metal  ribbon,  or  of  thicker 
rectangular  wire,  or  of  round  wire.  In  each  case,  the  inductance 
calculated  for  the  equivalent  coil,  whose  dimensions  are  measured 
by  the  method  about  to  be  treated,  will  generally  be  as  close  as 
i  per  cent  to  the  truth,  the  value  thus  computed  being  too  small. 

If  n  wires,  Fig.  184,  of  rectangular  cross  section  are  used,  whose 
width  in  the  direction  of  the  axis  is  w,  whose  thickness  is  t,  and 
whose  pitch,  measured  from  the  center  of  cross  section  of  one  turn 
to  the  corresponding  point  of  the  next  wire  is  D,  then  the  dimen- 
sions of  the  cross  section  of  the  equivalent  coil  are  to  be  taken  as 


b=w,  c  =  nD,  and  as  before  d  =  -y62  +  c2. 

The  mean  radius  of  the  equivalent  coil  is  to  be  taken  as  a  = 
Oi  +  /4(n—i)D,  the  distance  a^  being  one-half  of  the  distance  AB 
(see  Fig.  185)  measured  from  the  innermost  end  of  the  spiral 
across  the  center  of  the  spiral  to  the  opposite  point  of  the  inner- 
most turn. 

The  inductance  Lu  of  the  equivalent  coil  is  to  be  calculated 
using  the  above  dimensions  in  (158),  assuming  for  n  the  same 
number  of  turns  as  that  of  the  spiral. 


Radio  Instruments  and  Measurements 


261 


If  round  wire  is  employed,  the  same  method  is  used  for  obtain- 
ing the  mean  radius  a  and  the  dimension  c,  but  it  is  more  con- 
venient to  take  b  as  zero,  and  use  for  the  calculation  of  the  induc- 
tance of  the  equivalent  coil  the  special  form  of  (158)  which  follows 
when  6  is  placed  equal  to  zero. 


Sa 


80 


f  Sa      i 

0.01257  n*a   2.303  Iog10  —  —  - 


FlG.   184. — Sectional  -view  of  flat 
spiral  wound  with  metal  ribbon 


FIG.  185. — Side  -view  of  flat  spiral 


The  correction  for  cross  section  may,  in  each  case,  be  made  by 
subtracting  0.01257  na  (Al+51)  from  the  value  of  inductance  for 
the  equivalent  coil. 

For  round  wires  the  quantities  Ax  and  Bl  may  be  taken  as 
equal  to  A  and  B  in  the  Tables  1 1  and  12,  page  284,  just  as  in  the 
case  of  single-layer  coils  of  round  wire. 

In  the  case  of  wire  or  strip  of  rectangular  cross  section  the  matter 
is  more  complicated  on  account  of  the  two  dimensions  of  the  cross 
section. 

TV  t 

If  we  let  jj  —  v  and  JJ  =  T,  then  the  quantities  involved  in  the 
calculation  of  At  and  Bl  may  be  made  to  depend  on  these  two 


262  Circular  of  the  Bureau  of  Standards 

parameters  alone.     The  equations  are  then  with  sufficient  accu- 
racy: 

1  V+l  V+l  I     H      \ 

A!  =  log.  ;^.-  2.303  tag^j—  (161) 

|~n  —  I.        n  —  2  s        n  —  3  s  ,  J  s  "1 

#!=-2 512H S13  + -flu +  •  .  . +-Sn  (162) 

L    n  n  n  n     J 

in  which  612,  513,  etc.,  are  to  be  taken  from  Table  15,  page  285. 

Example. — For  a  spiral  of  38  turns,  wound  with  copper  ribbon 
whose  cross  sectional  dimensions  are  3/8  by  1/32  inch,  the  inner 
diameter  was  found  to  be  2at  =  10.3  cm  and  the  measured  pitch 
was  found  to  be  0.40  cm. 

The  dimensions  of  the  equivalent  coil  of  rectangular  cross  sec- 
tion-are, accordingly, 

6  =  3/8  inch  =  0.953  cm, 

a  =  ^  +  -^-37  (o-4)  =12.55, 


£  =  38  X  0.40  =  15.  2. 
For  this  coil  b/c  =  0.0627  which   gives  (Table    14)  ^=0.5604, 

d2  Sa 

^3  =  0.599,^=  1.472,  log.  -j-i.886. 

Hence  from  (158), 

Lu=  (0.01257)  (12.55)  (38)2  [1.015  (1.886)  -0.5604  +  0.055] 

=  323.3  microhenries. 
For  this  spiral  ^  =  2.38,  r  =  0.198 

7    •jfi 

A,  =  2.303  Iog10^fg  =0.270 


=  ~2     (0>028)  +    (0<OI3)  +    (0-007)  +    (0-°°4) 
+||  (0.003)  +||  (0.002)  +||  (0.002)4-^(0.001)+  .  •  -1 


=  -0.112,    ^ 

and  the  total  correction  is  (0.01257)  (38)  (12.55)  (0.159)  =o-95  M 
so  that  the  total  inductance  of  the  spiral  is  324.2  microhenries. 
The  measured  value  was  323.5. 

INDUCTANCE  OF  A  SQUARE  COIL 

Two  cases  present  themselves 

(a)  A  square  coil  wound  in  a  rectangular  cross  section. 

(6)  A  square  coil  wound  in  a  single  layer. 


Radio  Instruments  and  Measurements  263 

MULTIPLE-LAYER  SQUARE  COIL 

Let  a  be  the  side  of  the  square  measured  to  the  center  of  the 
rectangular  cross  section  which  has  sides  b  and  c,  and  let  n  be  the 
total  number  of  turns. 

Then 

Lu  =0.008  an2    2.303  Iog10  —^  +  0.2235-^  +0.726         (163) 
If  the  cross  section  is  a  square,  b  =  c,  this  becomes 

Lu  =  0.008  an2    2.303  Iog10 1 +  0.447 -+0.033  (l64) 

A  correction  for  the  insulating  space  between  the  wires  may  be 
calculated  by  equation  (159)  if  we  replace  0.01257  an  therein  by 


FIG.  1 86. — Multiple-layer 
square  coil  with  winding 
of  rectangular  cross  sec- 
tion 


FlG.  187. — Single-layer  square  coil 


0.008  an.     This  correction  is  additive,  but  will  be  negligible  unless 
the  insulating  space  between  the  wires  is  large. 

SINGLE-LAYER  SQUARE  COIL 

Let  a  =  the  side  of  the  square,  measured  to  the  center  of  the  wire 
n  =  number  of  turns 

D=  pitch  of  the  winding,  that  is,  the  distance  between  the 
center  of  one  wire  and  the  center  of  the  next  Fig.  187 


264  Circular  of  the  Bureau  of  Standards 

Then 

L0=  0.008  an2    2.303  Iog10  £  +  0.726  +  0.2231  - 

-0.008  an  [A  +B]  (165) 

in  which  A  and  B  are  constants  having  the  same  meaning  as  in 
(155)  to  be  taken  from  Tables  n  and  12,  if  the  wires  are  of  round 
cross  section.  If  the  wire  is  a  rectangular  strip  having  a  dimen- 
sion t  along  the  axis  of  the  coil  and  w  perpendicular  to  it,  calculate 
Lu  by  (163)  and  correct  for  cross  section  by  (161)  and  (162)  and 
Table  15,  using  0.008  an  (A^+B^. 

Example.  —  Suppose  a  square  coil,  100  cm  on  a  side,  wound  in  a 
single  layer  with  4  turns  of  round  wire,  o.i  cm  bare  diameter,  the 
winding  pitch  being  0.5  cm. 

In  this  case  n  =  4  d  =  o.i         6  =  4X0.5  =  2.0 

a  =  ioo        D=o.5 
The  main  term  in  formula  (165)  gives 

0.008X100X16    2.303  Iog10  --  [-0.726  +  0.004 
=  128  [4.  638  +0.726  +  0.004]  =  59.42  microhenries 

Entering  Tables  1  1  and  1  2  ,  page  284,  with  jj  =  -1-  =  0.2  and  n  =  4, 

A  =  -1.053 
B=     0.197 

sum=  —0.856 

0.008  an  [  —  0.856]=  —2.74  microhenries, 
so  that  Lu  =  59.42  +  2.  74  =  62.16. 

This  result  may  be  checked  by  computing  the  self-inductance 
Lt  of  a  single  turn  and  the  mutual  inductances  Mpq  of  the  indi- 
vidual turns,  and  summing  them  up. 

Thus  we  find 

4/^  =  22.65 


=   5-5Q 
62.18  microhenries. 

Formula  (165)  applies  only  when  the  length  b  is  small  com- 
pared with  the  side  of  the  square  a. 


Radio  Instruments  and  Measurements  265 

RECTANGULAR  COIL  OP  RECTANGULAR  CROSS  SECTION 

the  sides  of  the  rectangle  be  a  and  alt  the  dimensions  of  the 
cross  section  b  and  c,  and  the  number  of  turns  n,  g  =  -^ja2  +  a,2 

2aal         a 
Lu = 0.0092 1  (a  +  aj  n2    log  10  rr~ r —  *o£  10  (a  +  9) 

I—  *• 


(166) 


a  and  at  be  the  sides  of  the  rectangle,  D  the  pitch  of  the 
winding,  b  =  nD,  and  n  the  number  of  turns.     Then 


Correct  for  cross  section  by  (159)  for  round  wire. 

SINGLE-LAYER  RECTANGULAR  COIL 


FIG.  188. — Single-layer  rectangular  coil. 


L0  =  0.0092 1  (a  +  aj 


log 


n 


-0.447 


-  0-004  (a  +  Oi)  n  (A  +5) 


(167) 


where  A  and  B  are  to  be  taken  from  Tables  n  and  12,  if  the  coil 
is  wound  with  round  wire.  If  wound  with  strip,  take  b=nD  and 
c=  radial  thickness  of  strip.  Calculate  Lu  by  (166)  and  correct 
for  cross  section  by  (161),  (162),  and  Table  15. 


266 


Circular  of  the  Bureau  of  Standards 

FLAT  RECTANGULAR  COIL 


Let  a0  and  a'0  be  the  outside  dimensions  of  the  coil,  measured 
between  centers  of  the  wire,  D  the  pitch  of  the  winding,  meas- 
ured between  the  centers  of  adjacent  wires  (Fig.  189),  n  the  num- 
ber of  complete  turns,  d  the  diameter  of  the  bare  wire,  c=nD, 


=  Lu  -  0.004  n(a  +  aj  (A  +  B) 


2aa, 


</  =  diagonal  =  Ya2  +  a12,  a  =  a0—(n—i)D,  a1  =  a'0-(n-i)D. 
Then 

where 

Lu  =  o.oo92io  n2!  (a  +  ajlogio^^  — alog10(a  +  <7) 

]r        a  +  a  -i 

+  0.004  n\  29 ~+°-447  c     (*68) 

and  A  and  B  are  constants  to  be  taken  from  Tables  n  and  12 
for  round  wire.  If  the  coil  is  wound  with  rectangular  strip,  put 
b=  width  of  the  strip,  and  c  =  nD,  and  calculate  Lu  by  (166) 
using  for  A  andB  the  values^  and  Z^of  (161)  and  (162)  Table  15. 

FLAT  SQUARE  COIL 

If  a0  be  here  the  side  of  the  square,  measured  between  centers 
of  two  outside  wires,  and  a  =  a0—  (n—  i)D,  the  nomenclature 
being  as  in  the  previous  section, 


in  which 


FlG.  189. — Flat  square  coil. 

Lo=Lu- 0.008  n  a  (A 


Lu  =  0.008  n*a\  2.303  Iog10-  +  0.2235  ^  +  0.726  (169) 


Radio  Instruments  and  Measurements  267 

For  round  wire  the  constants  A  and  B  are  given  in  Tables  n 
and  12.  If  the  coil  is  wound  with  strip  proceed  as  for  rectangular 
flat  coils  of  strip,  above. 

Example.  —  A  coil  of  4  turns  of  0.22  cm.  stranded  wire  was  found 
to  have  a0  =  102  cm,  the  pitch  of  the  winding  being  D  =  2.25  cm. 

Here 

0  =  102-3x2.25=95.25 

c  =  4X2.  25=9.0 


.  25!  2.303 


Lu  =  0.008X16X95.  25   2.303  Iog10 

=  16X0.762  [2.359+0.021  +0.726]  =  37.87  iih 
For 

n  =  4  and  7^  =  —  —==0.098,  Tables  n  and  12  give 
D     2.25 

A  =  —  1.767,  and  B  =0.197 

the  correction  is  0.008x4X95.25   (  —  1.570)  =  —4.79^  so 
that  L0  =  37.  87  +4.  79  =42.  66  microhenries. 

The  measured  value,  unconnected  for  lead  wires  was  44.5 
microhenries. 

DOUBLE  FLAT  RECTANGULAR  COIL 

Such  a  coil  consists  of  two  similar  flat,  rectangular  coils,  such 
as  are  treated  in  the  preceding  sections,  placed  with  their  axes 
in  the  same  straight  line,  and  their  planes  at  a  distance  x  apart. 
The  two  sections  of  such  a  coil  may  be  used  either  singly,  or  in 
series,  or  in  parallel. 

The  general  method  of  treatment  is  to  obtain  the  inductance 
L!  of  the  single  sections  by  formula  (168)  or  (166),  as  described 
in  the  preceding  sections,  and  the  mutual  inductance  of  the  two 
sections,  as  shown  below. 

Then  when  used  in  series  L'  =  2(L1+M),  and  when  used  in 

parallel  L"  J*. 


To  obtain  the  mutual  inductance,  formula  (183)  or  (184)  for 
two  equal,  parallel  rectangles  or  squares,  multiplied  by  the  prod- 
uct of  the  number  of  turns  of  the  two,  should  be  used,  putting 
for  the  dimensions  of  the  rectangles  a  and  at  as  defined  under 
(168)  and  (169)  and  for  the  distance  D  in  (183)  or  (184)  a  mod- 
ified distance  r  given  by  the  expression 

r  =  kc,  c  =  nD,  (x/c  small) 


268  Circular  of  the  Bureau  of  Standards 

in  which 

2.303  Iog,,fc-2.3o3      log,.     +,£-5-a 


When  x  is  not  small  in  comparison  with  c,  r  will  have  to  be  cal- 
culated by  the  equation 

/  x^       ,  c  *\ 

I    o fo-n  -1 iZ  I 

ft  T     /  V*  \  I      *          1x111 

*     -                       I  /               *    \  .              x  -          ->\\C                 X  2  J 

•)    i    \       /     /»__\ 


logio  r  =  ~  Iog10  *  +  -(  i  -  7i  )  log! 

2\  c/  2.303 

When  the  distance  x  between  the,  planes  of  the  coils  is  chosen 
equal  to  the  pitch  D  of  their  windings,  the  calculation  of  their 
inductance,  when  joined  in  series,  may  be  obtained  in  a  simpler 
manner.  Putting  b  =  2D  and  ni  =  2n,  the  number  of  turns  of  the 
two  windings  in  series, 

a  b  +  c 

L  =0.008  n^a]  2.303  Iog10  7— — (-0.2235 1-0.726 

L  b+c  a  J 

+0.008  n^a  I  2.303  Iog10  j  +  0.153  J  (172) 

for  a  square  coil,  and 

V  =  0.0092 1  o  tt!2    (a  +  aj  Iog10  r— -1  -  a  Iog10  (a  +  g) 

(/  *  I    C 

I  tt  ~{~  di 

J  L  2  J 

+  0.004  ^i  («  +  Oi)  I  2-303  Iog10  ^  +  o.  1 53 J  ( 1 73) 

for  a  rectangular  coil 

<7  =  -\ja2  +  at2,  J  =  diameter  of  bare  wire. 

Example. — As  an  example  of  the  use  of  these  formulas,  take 
the  case  of  an  actual  coil  of  two  sections,  each  being  a  flat,  square 
coil  of  5  turns  of  0.12  cm  wire,  wound  with  a  pitch  of  .0  =  1.27 
cm,  the  distance  of  the  planes  of  the  coils  being  #  =  1.27  cm. 
The  length  of  a  side  of  the  outside  turn  was  101  cm. 

Putting  n  =  5,  a  =  101  -4X1. 27  =95.9,  0  =  5X1.27  =  6.35,  and 
d/D=o.i,  formula  (169)  gives  ^  =  66.28  +  6.14  =  72.42^,  for  a 
single  section. 


Radio  Instruments  and  Measurements  269 

To  obtain  the  mutual  inductance,  we  find  by  (170)  for 


S_1.27 

£"6.35 


=  0.2 


3     3, 


2.303  Iog10  k=2.  303X0.04  (-0.699)  +0.2  if  --  ^(0.04)  -^(o.ooi  6) 

=  —0.06444-0.6283  —  1.5—0.06  —  0.0001 
=  -0.9962 
Iog10fe=  -0.4326  =  1.5674 

k  =0.3693  and  r  =  0.3693  X  6.35  =  2.344 

Putting  this  value  of  r  in  place  of  D  in  (184)  with  0  =  95.9 
M  =0.008  X  5  X  5  [2.303  X  95-9  Iog1 

—  191.86  +  2.34    =56. 
For  the  two  coils  in  series,  then 


+  135-62 


.82 


L'  =  2(72.42  +56.82)  =258.5  ph 
and  for  the  parallel  arrangement 


The  inductance  of  the  coils  in  series  may  also  be  found  by 
putting  0  =  95-9*  b  =  6.i,5,  ^  =  2.54,  ^  =  10  in  (163)  and  (159)  and 
we  find  L  =  239.8  +  18.8  =  258.  6  ph  in  agreement  with  the  other 
method. 

69.  MUTUAL  INDUCTANCE 


The  following  formulas  for  mutual  inductance 
hold  strictly  only  for  low  frequencies.  In  gen- 
eral, however,  the  values  will  be  the  same  at  high 
frequencies. 


A 


TWO  PARALLEL  WIRES  OR  BARS  SIDE  BY  SIDE 

Let  /  =  length  of  each  wire  or  bar. 

D  =  distance  between  centers  of  the  wires. 


The  following  expression  is  exact  when  the  FIG.  igo.—Two 
wires  have  no  appreciable  cross  section,  but  is    kl  wires  side  bv side 
sufficiently  exact  even  when  the    cross    section  is  large  if  /  is 


270 


Circular  of  the  Bureau  of  Standards 


great  compared  with  D.    Within  these  limits  the  shape  is  im- 
material. 


=  0.002/      2.303 


I  2. 


D 


'] 


(174) 


=  o.oo2/    2.303  Iog10^  —  i  +-T    nearly.  (J75) 

TWO  WIRES  END  TO  END  WITH  THEIR  AXES  IN  LINE 

Let  the  lengths  of  the  two  wires  be  /  and  m,  their  radii  being 
supposed  to  be  small.     Then, 


l  +  m 
M  =  0.002  303    /Iog10—  -j— 


l  +  rnl 
-^- 


f     ,\ 
(176) 


TT\ 


FIG.  191. — Two  wires  end  to  end 
in  same  straight  line 


FIG.  192. — Two  wires  in  same  straight 
line  but  separated 


TWO  WIRES  WITH  THEIR  AXES  IN  THE  SAME  STRAIGHT  LINE  BUT  SEPARATED 

Let  their  lengths  be  /  and  m  and  the  distance  between  the  nearer 
ends  be  Z. 


M  =  0.002303  [(/  +  m  +  Z)  logxo  (/  +  m  +  Z)  +  Z  Iog10  Z 
-  (/  +  Z)  Iog10  (/  +  Z)  -  (m  +Z)  loglo  (m  +  Z)] 


d77) 


Radio  Instruments  and  Measurements 


271 


TWO  WIRES  WITH  AXES  IN  PARALLEL  LINES 

If  AD,  AD',  AC,  AC',  etc.,  represent  the  distances  shown  in 
the  figure,  the  general  formula  is 

Dl ^ 


C' 

&' 
i 


m 


B 


FIG.  193. — Two  wires  with  axis  in  parallel  lines 


r;1       <AD+ADf     AC-AC'\ 
0.001  151  [l^<>{AD-AD>XAc+AC'\ 


AD  +  AD'    BD-BD' 


iAD+AD'     AC-AC'     BD-BD'     BC+BC'\~] 
1°8">\AD-AD'XAC+AC'XBD+BD'XBC-BC'\] 

-o.ooi  (AD-AC-BD+BC) 


the  distances  being  AD'  =  l  +  m  +  Z,  AD  =  -\lx2  +  (l  +  m  +  Z)2,  etc. 
This  formula  holds  for  Z=o,  but  not  when  one  wire  overlaps  on 
the  other. 

When  they  overlap,  as  in  Fig.  194, 

M  =  M1(34  +  M23  +  M24  (179) 

in  which  M1)34  is  to  be  calculated  by  the  general  formula,  using 
Z  =o  and  putting  the  segment  PV  for  /  and  ST  for  m,  while  for 
M,4  the  length  VR  is  put  for  /  and  WT  for  m  with  Z=o.  The 

35601°— 18 IS 


272 


Circular  of  the  Bureau  of  Standards 


mutual  inductance  M23  of  the  overlapping  portions  is  obtained 
by  (174)- 

T 


R 


V 


w 
S 


FIG.  194. — Two  wires  with  axis  in  parallel 
lines;  a  particular  case  of  Fig.  193 

Special  Cases. — For  the  case  shown  in  Fig.  195 
M  =  0.001 


-D 


D- 


(180) 


FIG.  195. — Two  wires  with  axes  in  parallel    FIG.  196. — Two  wires  with  axes  in  parallel  lines, 
lines;  another  particular  case  of  Fig.  193     -with  one  end  of  each  on  the  same  perpendicular 

and  for  the  wires  of  Fig.  196 
M=o.oo,  [4.605,  logl 

llblj 


Radio  Instruments  and  Measurements  273 

MUTUAL  INDUCTANCE  OF  TWO  PARALLEL  SYMMETRICALLY  PLACED  WIRES 


2* 

t 


2JI, 


FIG.  197 . — Two  parallel  symmetrically 
placed  wires 

Putting  for  the  lengths  of  the  two  wires  2/  and  2/j  (2/  the  shorter) 
and  for  their  distance  apart  D 


M 


=  0.002  I   2.303(2/)log10| l       V  D      l 


(182) 


TWO  EQUAL  PARALLEL  RECTANGLES 


Let  a  and  at  be  the  sides  of  the  rectangles  and  D  the  distance 
between  their  planes,  the  centers  of  the  rectangles  being  in  the 
same  line,  perpendicular  to  these  planes 


)    a 


,- 

M  =  0.009210    a log 8.     2 

L  [a  +  V°  +  ai 


^ 
J 


-  -Ja2+D*  -  Jaf+D*  +  D] 


(183) 


274 


Circular  of  the  Bureau  of  Standards 


TWO  EQUAL  PARALLEL  SQUARES 

If  a  is  the  side  of  each  square  and  D  is  the  distance  between 
their  planes,  then  the  preceding  formula  becomes 


:is4) 


+  0.008 


+  D2  -  2    a2  +D2  +D] 


MUTUAL  INDUCTANCE   OF  TWO  RECTANGLES   IN  THE  SAME  PLANE  WITH   THEIR 

SIDES  PARALLEL 


M  =  2(MW 


+M27)  -  2(M18  +M25  +M33  +M47)      (185) 


,.6 


'8 


FIG.  198. — Two  rectangles  in  the  same  plane 
•with  their  sides  parallel 

the  separate  mutual  inductances  being  calculated  by  formula 
(182),  if  the  sides  are  symmetrically  placed,  and  by  (182)  and 
(178)  if  that  is  not  the  case. 

If  the  rectangles  have  a  common  center  M18  =  M38,  M45  =  M,7, 
M18  =  M3e,  M25=M47  and  for  the  case  of  concentric  squares,  we 
have 


=  4(M18-M18) 


(186) 


TWO  PARALLEL  COAXIAL  CIRCLES 


This  is  an  important  case  because  of  its  applicability  in  calcu- 
lating the  mutual  inductances  of  coils  (see  below) . 
Let  a  =  the  smaller  radius  (Fig.  199). 
A  =the  larger  radius. 

D  =  the  distance  between  the  planes  of  the  circles. 
Then 


- 


Radio  Instruments  and  Measurements 


275 


(i87) 
where  F  may  be  obtained  by  interpolation  in  Table  16  for  the 

calculated  value  of  — •  ~ -'"? 


fx  =  the  longest  distance  between 
the  circumferences. 

r2  =  the  shortest  distance  between 
the  circumferences. 


TWO  COAXIAL  CIRCULAR  COILS  OF  RECTANGULAR 
CROSS  SECTION 

If  the  coil  windings  are  of  square,  or 
nearly  square,  cross  section,  a  first  ap- 
proximation to  the  mutual  inductance  is 

(188) 


where  n^  and  n2  are  the  number  of  turns 
on  the  two  coils  and  M0  is  the  mutual 
inductance  of  two  coaxial  circles,  one 
located  at  the  center  of  the  cross  section 
of  one  of  the  coils  and  the  other  at  the 
center  of  the  cross  section  of  the  other. 
Thus,  if 


,Vv 

/ 

1 

r 

\ 

\ 

\ 

\ 

A 

\ 

\ 

a 

\ 

\ 

r«—  V-D  —  ^ 

\ 

\ 

\ 

\ 

\ 

\ 

\  ' 

\ 

\ 

1 

\ 

1 

1 

i 

,                                   V 

1 

i 

\ 

1 

\ 

FIG.  199. — Cross  sections  of  two 
parallel  coaxial  circles 


FIG.  200. — Two  paral- 
lel coaxial  coils  with 
windings  of  rectangu- 
lar cross  sections 


a  =  mean  radius  of  one  coil,  measured  from 
the  axis  to  the  center  of  cross  section, 

A  =  mean  radius,  similarly  measured,  of  the 
other  coil, 

D  =  distance  between  the  planes  passed 
through  the  centers  of  cross  section  of 
the  coils,  perpendicular  to  their  com- 
mon axis  (Fig.  200). 

the  value  M0  will  be  computed  by  formula 
(187)  and  Table  16,  using  the  values  of  a,  A, 
and  D,  just  defined. 

If  the  cross  sections  of  the  windings  are 
square,  this  value  will  not  be  more  than  a 
few  parts  in  a  thousand  in  error,  even  with 
relatively  large  cross  sectional  dimensions, 


except  when  the  coils  are  close  together. 


276  Circular  of  the  Bureau  of  Standards 

A  more  accurate  value  for  coils  of  square  cross  section  may  be 
obtained  by  supposing  the  two  parallel  circles  to  remain  at  the 
distance  D,  but  to  have  radii 


where  6j  and  b2  are  the  dimensions  of  the  square  cross  sections 
corresponding  to  the  coils  of  mean  radius  a  and  A,  respectively. 

When  the  correction  factors  in  (189)  are  only  a  few  parts  in 
1000,  the  values  of.rjrlt  and  hence  F,  are  very  little  affected,  and 
the  fractional  correction  to  the  mutual  inductance,  to  allow  for 
the  cross  sections,  is  approximately  equal  to  the  geometric  mean 
of  the  fractional  corrections  to  a  and  A  ,  so  that  an  estimate  of  the 
magnitude  of  the  correction  to  the  mutual  inductance  may  be 
gained  with  little  labor. 

With  rectangular  cross  sections  the  error  from  the  assumption 
that  the  coils  may  be  replaced  by  equivalent  filaments  at  the 
center  of  the  cross  section  is  more  important  than  in  the  case  of 
coils  of  square  cross  section  and  rapidly  increases  as  the  axial 
dimension  of  one  or  both  of  the  cross  sections  is  increased,  in  rela- 
tion to  the  distance  D  between  the  median  planes.  The  error 
may,  easily,  be  as  great  as  i  per  cent  or  more  in  practical  cases. 

An  estimate  of  the  magnitude  of  the  error,  in  any  case,  may  be 
made  by  dividing  the  coils  up  into  two  or  more  sections  of,  as 
nearly  as  possible,  square  cross  section,  and  assuming  that  each 
portion  of  the  coil  may  be  replaced  by  a  circular  filament  at  the 
center  of  its  cross  section. 

Suppose  that  coil  A  is  divided  into  two  equal  parts,  and  replaced 
by  two  filaments  i,  2,  while  coil  B  is  likewise  replaced  by  two 
filaments  3,  4,  then,  assuming  that  each  filament  is  associated 
with  a  number  of  turns  which  is  the  same  fraction  of  the  whole 
number  of  turns  in  the  coil  as  the  area  of  the  section  is  to  the 
whole  cross  sectional  area  (one-half  in  this  case)  we  have 

M^  2s  M,3+"'B*  Mu+**  Af,  +  ?-!=!  Ma 

4  4  4  (190) 


in  which  M13  is  the  mutual  inductance  of  the  two  circular  filaments 
i  and  3,  etc. 


Radio  Instruments  and  Measurements 


277 


For  a  discussion  of  more  accurate  methods  for  correcting  for 
the  cross  section  of  coils,  the  reader  is  referred  to  Bulletin, 
Bureau  of  Standards,  8,  pages  33-43;  1912. 

If  the  coils  are  of  the  nature  of  solenoids  of  few  layers,  it  is 
best  to  use  the  formulas  for  the  mutual  inductance  of  coaxial 
solenoids  given  in  the  next  section. 

Example. — Suppose  two  coils  of  square  cross  section  2  cm  on 
a  side,  the  radii  being,  a  =  20,  A  =  25,  and  the  distance  between 
their  median  planes  being  D  =  io  cm  (Fig.  201). 
Further,  suppose  that  one  coil  has  100  turns  and 
the  other  500. 
Then 


'0.24253 


From  Table  16  we  find,  corresponding  to  this 
value  of  — > 
F  =  o.oi  1 1 3.     Therefore,  from  (187) 

M0  =0.01 1 13-^25X20  =  0.2489^ 

and 

M  =  n^Mo  =  100  X  500  X  0.2489 
=  12  445  microhenries 
=  0.012445  henry. 
If  we  take  account  of  the  cross  sections  we  have  from  (189) 


=2°  (1-00042) 


FIG.  201. — Exam- 
ple of  two  paral- 
lel coaxial  coils 
with  windings 
of  rectangular 
cross  section 


so  that  the  correction  factor  to  the  mutual  inductance  will  be  of 
the  order  of  about  1/1.00042  X  1.00027,  or  the  mutual  inductance 
should  be  increased  by  about  3.5  parts  in  10  ooo  only. 

Example.  —  Fig.  202  shows  two  coils  of  rectangular  cross  section. 
For  coil  P,  a  =  20,  61  =  2,  ^  =  3,  7^  =  600.  For  coil  Q,  A  =25,  62  =  4» 
c2  =  i,  n2  =  400  and  D  =  io.  If,  first,  we  replace  each  coil  by  a 


278 


Circular  of  the  Bureau  of  Standards 


circular  filament  at  the  center  of  its  cross  section,  we  have  the 
1    same  value  of  M0  as  in  the  previous  example,  and 

M  =  600  X  400  X  0.2489  microhenries. 


More  precise  formulas,  involving  a  good  deal  of 
computation,  show  that  the  true  value  is 


M  =  600  x  400  x  o.  249844, 


\ 


so  that  the  approximate  value  is  about  3.8  parts  in 
looo  too  small. 

Each  coil  is  then  subdivided  into  two  sections 
and  filaments  p,  q,r,  s,  imagined  to  pass  through 
Another  ^    center  of  cross  section  of  each  of  these  subdivi- 

example  of  rig. 

200  sions.     The  data  for  these  filaments  are  as  follows : 


Radius 

Filaments 

a 

A 

D 

rz/ri 

F 

p  19.  25 

pr 

19.25 

25 

9 

0.  2365 

0.  01140 

q  20.  75 

ps 

19.25 

25 

11 

.2722 

.  009872 

r  25 

V 

20.75 

25 

9 

.2135 

.  01255 

s  25 

qs 

20.75 

25 

11 

.2506 

.  01077 

We  find  then 

[0.2501+0.2166+0.2858+0.2452! 


M =600X400 


4 


=600X400X0.24942 


a  result  which  is  1.7  in  1000  too  small. 

The  increase  in  accuracy  is  hardly  commensurate  with  the 
increased  labor. 

MUTUAL  INDUCTANCE  OF  COAXIAL  SOLENOIDS  NOT  CONCENTRIC 

Gray's  formula,  given  for  this  case,  supposes  that  each  coil 
approximates  the  condition  of  a  continuous  thin  winding,  that  is, 

a  current  sheet. 

1x_ ^ 

!*""^-.  n 


~n. 
a, 

~*^f 


K-- 


•_  _  y  _•   ^j 

P- *1  1 

FIG.  203. — Coaxial  solenoids  not  concentric 


Radio  Instruments  and  Measurements 


279 


Let  a  =  the  smaller  radius,  measured  from  the  axis  of  the  coil 

to  the  center  of  the  wire 

A  =the  larger  radius,  measured  in  the  same  way 
2/  =  length  of  the  coil  of  smaller  radius  =  number  of  turns 

times  the  pitch  of  winding 
2%  =  length  of  the  coil  of  larger  radius,  measured  in  the  same 

way 
nx  and  n2  =  total  number  of  turns  on  the  two  coils 

D  =  axial  distance  between  centers  of  the  coils 
x-i  =  D  —  x  t\  = 

Then 

a2  A2* 


M  =  0.0098  70 


&  +  Ksk3  +  K6ks 


in  which 


(191) 


A*T  x^        x,-\      za/        x22 

=  -  TL^(3  -  fa)  -  f?  (3  -  ^ 

(z 
^- 
2 


a~ 

This  formula  is  most  accurate  for  short  coils  with  relatively 
great  distance  between  them.  In  the  case  of  long  coils  it  is  some- 
times necessary  to  subdivide  the  coil  into  two  or  more  parts. 
The  mutual  inductance  of  each  of  these  parts  on  the  other  coil 
having  been  found,  the  total  mutual  inductance  is  obtained  by 
adding  these  values. 

Example.  — 


•20.53- 


£7.i&- 


4   ' 

6.44 

t 

>-*-7.2-* 

4.455 

* 

FIG.  204. — Example  of  coaxial  solenoids  not  concentric 
2X  =  20.55  .4=6.44  ^i  =  I5 

2/  =27.38  a  =4.435  ™2  =  75 

Distance  between  the  adjacent  ends  of  the  two  solenoids  =  7.2  cm. 


280  Circular  of  the  Bureau  of  Standards 

Then 

*!  =  20.89          k1Kl  =0.04294 
£2  =  4! -44          k3K3=  .01827 

=  .00519 


0.06640 

/a2A2n.n«\ 
and  M  =  0.009870! M  10.06640  =  i  .069  microhenries 

log  0.009870  =  3.99432 

2  log  a  =1.29378  log  2^  =  1.31281 

2  log  A          =1.61778  log  2/ =1.43743 

lognjw,          =3.05115 

log  0.06640   =2.82217  2.75024 

2.77920 
2.75024 


0.02896  =  log  M 

Dividing  the  longer  coil  into  two  sections  C  and  D  of  37  and  38 
turns,  respectively,  and  repeating  the  calculation  for  the  mutual 
inductance  of  these  sections  on  the  other  coil  R  (Fig.  204) , 

For  MRC  For  MRD 

klKl=  0.04889  k1K1=  0.01155 

k3K3  =   .00652  k3K3=   .00061 
k5K5=   .00005 


0.05546 


0.01216 


and  M  =  MHO  +  MRD  =  0.891 7  4- o.i 956  =  1.087  M- 

Further  subdivision  showed  that  this  last  value  is  not  in  error 
by  more  than  5  parts  in  10  ooo. 

The  criterion  as  to  the  necessity  of  subdivision  is  the  rapidity 
with  which  the  terms  kJK-i,  k3K3,  etc.,  fall  off  in  value.  In  the 
first  case  k7K7  and  k^K9  are  not  negligible.  The  expressions  for 
these  quantities  are  not  here  given  because  they  are  laborious  to 
calculate,  and  it  is  easier  to  obtain  the  value  of  the  mutual  induc- 
tance by  the  subdivision  method. 

COAXIAL,  CONCENTRIC  SOLENOIDS  (OUTER  COIL  THE  LONGER) 

The  formula  here  given  holds,  strictly,  only  for  current  sheets. 
The  lengths  of  the  coils  should  be  taken  as  equal  to  the  number 
of  turns  times  the  pitch  of  the  winding  in  each  case.  Then  the 


Radio  Instruments  and  Measurements  281 

mutual  inductance  of  the  current  sheets  is  not  appreciably  differ- 
ent from  that  of  the  coils. 

Let  a  =  smaller  radius  .,  / 

A  =  larger  radius 
2#  =  equivalent 
length  of  outer  coil 
2/  =  equi  valent 
length  of  inner  coil 


g  =  ix*+A2  =  diag- 
onal. 


FIG.  205.  —  Coaxial  concentric  solenoids,  outer  coil 
begin  longer 


This  formula  is  more  accurate,  the  shorter  the  coils  and  the  greater 
the  difference  of  their  radii,  but  in  most  practical  cases  the  accu- 
racy is  ample.  In  many  cases  the  second  term  in  (192)  is  negli- 
gible, and  it  is  a  good  plan  to  make  a  preliminary  rough  calcula- 
tion of  this  term  to  see  whether  it  will  need  to  be  considered.  In 
the  case  of  long  coils,  and  of  coils  of  nearly  equal  radii,  the  terms 
neglected  in  this  formula  may  be  as  great  as  i  per  cent.  A  crite- 

o?A2 
rion  of  rapid  convergence  is,  in  general,  the  smallness  of  —  j->  but 

(12\ 
3-4—3  j  and  the  corresponding 

coefficients  of  terms  neglected  in  (192)  may  in  some  cases  modify 
this  condition  for  rapid  convergence  materially. 

Example.  — 


=  300      n2  =  200 


y 
0.01974—  !-?=  1  198.5 

M  =  1  198.5  (i  +  .001  15)  =  1  199.9  microhenries. 

For  the  case,  however,  where 

2%  =  30  a  =  2          nl  =  300 

2/  =  24          A  =  5          n2  =  960 


282 


Circular  of  the  Bureau  of  Standards 


a2  A2        i  /  72\ 

although  the  value  of  -    r=-     -  only,  the  coefficient  (3-4  -=  1 

9        5°°°  \          a2/ 


=  141,  (the  length  of  the  coil  is  great  compared  with  its  radius)  so 

a2  A2 

is   —0.0282,  and   investigation  of  the  corn- 


that  the  term  in 


plete  formula  shows  that  the  succeeding  terms  are  —0.0127  and 
—0.0048,  so  that  their  neglect  will  give  an  error  of  over  1.5  per 
cent.  (For  precision  calculations  see  Bull.,  Bureau  of  Standards, 
8,  pp.  61-64,  !9i2,  for  the  complete  formula.) 

CONCENTRIC  COAXIAL  SOLENOIDS  (OUTER  COIL  THE  SHORTER) 


I 

% 

/ 

•o 

t 

-<  r^  ' 
/ 

y                                ) 

-  6 

"Z  -  —  i 

»— 

FIG.  206.  —  Coaxial  concentric  solenoids,  outer  coil  bing  shorter 

In  this  case  we  have  to  put  g  =  ^ 


and  the  formula  is 


('93) 
which  is  rapidly  convergent  in  most  cases. 

70.  TABLES  FOR  INDUCTANCE  CALCULATIONS 

TABLE  8.—  Values  of  d  in  Formulas  (132),  (134),  (137),  (138),  (140),  (141),  (142),  and 
(147),  for  Calculating  Inductance  of  Straight  Wires  at  Any  Frequency 


I 

£ 

z 

i 

0 

0.250 

12.0 

0.059 

0.5 

.250 

14.0 

.050 

1.0 

.249 

16.0 

.044 

1.5 

.247 

18.0 

.039 

2.0 

.240 

20.0 

.035 

2.5 

0.228 

25.0 

0.028 

3.0 

.211 

30.  0                .  024 

3.5 

.191 

40.  0                .  0175 

4.0 

.1715 

50.0 

.014 

4.5 

.154 

60.0 

.012 

5.0 

0.139 

70.0 

0.010 

6.0 

.116 

80.0 

.009 

7.0 

.100 

90.0 

.008 

8.0 

.088 

100.0 

.007 

9.0 

.078 

00 

.000 

10.0 

.070 

Radio  Instruments  and  Measurements  283 

TABLE  9.— Constants  P  and  Q  in  Formulas  (141),  (142),  (144),  and  (145) 


2A 
I 

P 

I 

2A 

Q 

2A 
( 

P 

I 

2A 

Q 

0 

0 

0 

.0000 

0.6 

0.5136 

0.6 

1.2918 

0.1 

0.0975 

0.1 

.0499 

.7 

.5840 

.7 

1.  3373 

.2 

.1900 

.2 

.0997 

.8 

.6507 

.8 

1.  3819 

.3 

.2778 

.3 

.1489 

.9 

.7139 

.9 

1.  4251 

.4 

.3608 

.4 

.1975 

1.0 

.7740 

1.0 

1.  4672 

.5 

.4393 

.5 

.2452 

TABLE  10.— Values  of  K  for  Use  in  Formula  (153) 


Diameter 

Diameter 

£• 

Diameter 

E< 

Length 

K 

Difference 

Length 

Difference 

Length 

jK 

DiSerenca 

0.00 

1.0000 

-0.  0209 

2.00 

0.  5255 

-0.0118 

7.00 

0.  2584 

-0.0047 

.05 

.9791 

203 

2.10 

.5137 

112 

7.20 

.2537 

45 

.10 

.9588 

197 

2.20 

.5025 

107 

7.40 

.2491 

43 

.15 

.9391 

190 

2.30 

.4918 

102 

7.60 

.2448 

42 

.20 

.9201 

185 

2.40 

.4516 

97 

7.80 

.2406 

40 

0.25 

0.  9016 

-0.0178 

2.50 

0.  4719 

-0.0093 

8.00 

0.2366 

-0.0094 

.30 

.8838 

173 

2.60 

.4626 

89 

8.50 

.2272 

86 

.35 

.8665 

167 

2.70 

.4537 

85 

9.00 

.2185 

79 

.40 

.8499 

162 

2.80 

.4452 

82 

9.50 

.2106 

73 

.45 

.8337 

156 

2.90 

.  4370 

78 

10.00 

.2033 

0.50 

0.  8181 

-0.0150 

3.00 

0.  4292 

-0.0075 

10.0 

0.2033 

-0.0133 

.55 

.8031 

146 

3.10 

.4217 

72 

11.0 

.1*03 

113 

.60 

.7885 

140 

3.20 

.4145 

•   70 

12.0 

.1790 

98 

.65 

.7745 

136 

3.30 

.4075. 

67 

13.0 

.1692 

87 

.70 

.7609 

131 

3.40 

.4000 

G4 

14.0 

.1605 

78 

0.75 

0.  7478 

-0.0127 

3.50 

0.  3944 

-0.0062 

15.0 

0.  1527 

-0.0070 

.80 

-7351 

123 

3.60 

.3882 

60 

16.0 

.1457 

63 

.85 

.7228 

118 

3.70 

.3822 

58 

17.0 

.1394 

58 

.90 

.7110 

115 

3.80 

.3764 

56 

18.0 

.1336 

52 

.95 

.6995 

111 

3.90 

.  3708 

54 

19.0 

.1284 

43 

1.00 

0.  6884 

-0.0107 

4.00 

0.  3654 

-0.  0052 

20.0 

0.  1236 

-0.0085 

1.05 

.6777 

104 

4.10 

.3602 

51 

22.0 

.1151 

73 

1.10 

.6673 

100 

4.20 

.3551 

49 

24.0 

.1078 

63 

1.15 

.6573 

98 

4.30 

.3502 

47 

26.0 

.1015 

56 

1.20 

.6475 

94 

4.40 

.3455 

46 

28.0 

.0959 

49 

1.25 

0.  6381 

-0.0091 

4.50 

0.3409 

-0.0045 

30.0 

0.0910 

-0.0102 

1.30 

.6290 

89 

4.60 

.3364 

43 

35.0 

.0808 

80 

1.35 

.6201 

86 

4.70 

.3321 

42 

40.0 

.0728 

64 

1.40 

.6115 

84 

4.80 

.3279 

41 

•  45.0 

.0664 

53 

1.45 

.6031 

81 

4.90 

.3238 

40 

50.0 

.0611 

43 

1.50 

0.  5950 

-0.  0079 

5.00 

0.  3198 

-0.0076 

60.0 

0.0528 

-0.0061 

1.55 

.5871 

76 

5.20 

.3122 

72 

70.0 

.0467 

48 

1.60 

.5795 

74 

5.40 

.3050 

69 

80.0 

.0419 

38 

1.65 

.5721 

72 

5.60 

.2981 

65 

90.0 

.0381 

31 

1.70 

.5649 

70 

5.80 

.2916 

62 

100.0 

.0350 

1.75 

0.  5579 

-0.0068 

6.00 

0.  2854 

-0.  0059 

1.80 

.5511' 

67 

6.20 

.2795 

56 

1.85 

.5444 

65 

6.40 

.2739 

54 

1.90 

.5379 

63 

6.60 

.2685 

52 

1.95 

.5316 

61 

6.80 

2633 

49 

284  Circular  of  the  Bureau  of  Standards 

TABLE  11.— Values  of  Correction  Term  A  in  Formulas  (155),  (165),  (168),  and  (169) 


d 

~D 

A 

Difference 

d 
D 

A 

Difference 

d 
D 

A 

Difference 

1.00 

0.557 

-0.  051 

0.40 

-0.359 

-0.052 

0.15 

-1.  340 

-0.069 

0.95 

.506 

54 

.38 

.411 

54 

.14 

1.409 

74 

.90 

.452 

57 

.36 

.465 

57 

.13 

1.483 

80 

.85 

.394 

61 

.34 

.522 

61 

.12 

1.563 

87 

.80 

.334 

65 

.32 

.583 

64 

.11 

1.650 

96 

0.75 

0.269 

-0.069 

0.30 

-0.  647 

-0.069  - 

0.10 

-1.746 

-0.  105 

.70 

.200 

74 

.28 

.716 

74 

.09 

1.851 

.118 

.65 

.126 

80 

.26 

.790 

80 

.08 

1.969 

.133 

.60 

.046 

87 

.24 

.870 

87 

.07 

2.102 

.154 

.55 

-  .041 

95 

.22 

.957 

96 

.06 

2.256 

.173 

0.50 

-0.136 

-0.041 

0.20 

-1.053 

-0.  051 

0.05 

-2.  439 

-0.223 

.48 

.177 

43 

.19 

1.104 

54 

.04 

2.662 

.288 

.46 

.220 

44 

.18 

1.158 

57 

.03 

2.950 

.405 

.44 

.264 

47 

.17 

1.215 

61 

.02 

3.355 

.693 

.42 

.311 

48 

.16 

1.276 

64 

.01 

4.048 

TABLE  12.— Values  of  Correction  B  in  Formulas  (155),  (165),  and  (169) 


Number  of 
turns,  7i 

B 

Number  of 
turns,  n 

B 

1 

0.000 

40 

0.315 

2 

.114 

45 

.317 

3 

.166 

50 

.319 

4 

..197 

60 

.322 

5 

.218 

70 

.324 

6 

0.233 

80 

0.326 

7 

.244 

90 

.327 

8 

.253 

100 

.328 

9 

.260 

150 

.331 

10 

.266 

200 

.333 

15 

0.286 

300 

0.334 

20 

.297 

400 

.335 

25 

.304 

500 

.336 

30 

.308 

700 

.336 

35 

.312 

1000 

.336 

TABLE  13.— Values  of  Be  for  Use  in  Formula  (156) 


b 

c 

B. 

b 

C 

B, 

1 

0.0000 

16 

0.  3017 

2 

.1202 

17 

.3041 

3 

.1753 

18 

.3062 

4 

.2076 

19 

.3082 

5 

.2292 

20 

.3099 

6 

0.  2446 

21 

0.  3116 

7 

.2563 

22 

.3131 

8 

.2656 

23 

.3145 

9 

.2730 

24 

.3157 

10 

.2792 

25 

.3169 

11 

0.2844 

26 

0.3180 

12 

.2888 

27 

.3190 

13 

.2927 

28 

.3200 

14 

.2961 

29 

.3209 

15 

.2991 

30 

.3218 

Radio  Instruments  and  Measurements 
TABLE  14.— Constants  Used  in  Formulas  (157)  and  (1S8) 


b/c  or  c/6 

n 

Difference 

C/6 

yi. 

Difference 

6/c 

V3 

Differ- 
ence 

0 
0  025 

0.5000 
.5253 

0.0253 
237 

0 

0.125 

0.002 

0 

0.597 

0.002 

'.05 

.5490 

434 

0.05 

.127 

5 

0.05 

.599 

3 

.10 

.5924 

386 

.10 

.132 

10 

.10 

.602 

6 

0.15 

0.6310 

0.0342 

0.15 

0.142 

0.013 

0.15 

0.608 

0.007 

.20 

.6652 

301 

.20 

.155 

16 

.20 

.615 

9 

.25 

.6953 

266 

.25 

.171 

20 

.25 

.624 

9 

.30 

.7217 

230 

.30 

.192 

23 

.30 

.633 

10 

0.35 

0.  7447 

0.0198 

0.35 

0.215 

0.027 

0.35 

0.643 

0.011 

.40 

.7645 

171 

.40 

.242 

31 

.40 

.654 

11 

.45 

.7816 

144 

.45 

.273 

34 

.45 

.665 

12 

.50 

.7960 

121 

.50 

.307 

37 

.50 

.677 

13 

0.55 

0.8081 

0.0101 

0.55 

0.344 

0.040 

0.55 

0.690 

0.012 

.60 

.8182 

83 

.60 

.384 

43 

.60 

.702 

13 

.65 

.8265 

66 

.65 

.427 

47 

.65 

.715 

14 

.70 

.8331 

52 

.70 

.474 

49 

.70 

.729 

13 

0.75 

0.  8383 

0.0039 

0.75 

0.523 

0.053 

0.75 

0.742 

0.014 

.80 

.8422 

29 

.80 

.576 

56 

.80 

.756 

15 

.85 

.8451 

19 

.85 

.632 

59 

.85 

.771 

15 

.90 

.8470 

10 

.90 

.690 

62 

.90 

.786 

15 

0.95 

0.8480 

0.0003 

0.95 

0.752 

0.064 

0.95 

0.801 

0.015 

1.00 

.8483 

1.00 

.816 

1.00 

.816 

TABLE  15.— Values  of  Constants  in  Formula  (162) 


t> 

Values  of  £12 

V 

Values  of  £13 

T=0 

0.1 

0.3 

0.5 

0.7 

0.9 

T=0 

0.3 

0.6 

0.9 

0 

0.114 

0.113 

0.106 

0.092 

0.068 

0.030 

0 

0.022 

0.020 

0.014 

0.004 

0.5 

.090 

.089 

.083 

.070 

.049 

.020 

0.5 

021 

.018 

.014 

.004 

1.0 

.064 

.064 

.059 

.050 

.034 

.013 

1.0 

019 

.018 

.013 

.004 

1.5 

.047 

.046 

.043 

.036 

.025 

.009 

2.0 

015 

.015 

.010 

.003 

2.0 

.035 

.035 

.032 

.027 

.018 

.007 

4.0 

008 

.008 

.005 

.002 

3.0 

.022 

.022 

.020 

.017 

.011 

.004 

6.0 

005 

.005 

.004 

.001 

4.0 

.015 

.015 

.014 

.012 

.008 

.003 

10.0 

003 

.003 

.002 

.005 

6.0 

.008 

.008 

.008 

.006 

.004 

.002 

8.0 

.006 

.006 

.005 

.004 

.003 

.001 

10.0 

.004 

.004 

.004 

.003 

.002 

.001 

V 

Values  of  in 

V 

Values  of  5is 

r=0               0.3 

0.6 

0.9 

T=0 

0.1 

0.5 

0.9 

0 

0.  009            0.  009 

O.*006 

0.002 

0 

0.005 

0.005 

0.004 

0.001 

1 

.009              .008 

.006 

.002 

5 

.003 

.003 

.002 

.001 

3 

.007              .006 

.004 

.001 

10 

.002 

.002 

.001 

.000 

5 

.  004              .  004 

.003 

.001 

10 

.002              .002 

.001 

.000 

V 

Values  of  «ie 

V 

Values  of  617 

V 

Values  of  &w 

T=0 

and  0.1 

0.5 

0.9 

T=0 

and  0. 

0.5 

0.9 

T=Q 

and  0.1 

0.5 

0.9 

0 

0.003 

0.003 

0.001 

0 

0.002 

0.002 

0.001 

0 

0.002 

0.001 

0.000 

5 

.002 

.002 

.000 

5 

.002 

.001 

.000 

5 

.001 

.001 

.000 

10 

.001 

.001 

.000 

10 

.001 

.001 

.000 

10 

.  001         .  001 

.000 

NOTE. — The  maximum  values  of  all  further  values  of  the  i's  are  o.ooi  or  less. 


286 


Circular  of  the  Bureau  of  Standards 


TABLE  16.— Values  of  F  in  Formula  (187)  for  the  Calculation  of  the  Mutual  Inductance 

of  Coaxial  Circles 


rj/ri 

F 

Difference 

r2/ri 

F 

Difference 

r«/ri 

F 

Difference 

0 

00 

0.010 

0.  05016 

-0.  00120 

0.30 

0.  008844 

—0.000341 

0.80 

0.  0007345 

-0.  0000504 

-.011 

4897 

109 

.31 

8503 

328 

.81 

6741 

579 

.012 

4787 

100 

.32 

8175 

314 

.82 

6162 

555 

.33 

7861 

302 

.83 

5607 

531 

0.013 

4687 

-0.  00093 

.34 

7559 

290 

.84 

5076  ' 

507 

.014 

4594 

87 

.015 

4507 

81 

0.35 

0.  007269 

-0.000280 

0.85 

0.0004569 

-0.0000484 

.016 

4426 

148 

.36 

6989 

270 

.86 

4085 

460 

.018 

4278 

132 

.37 

6720 

260 

.87 

3625 

437 

.38 

6460 

249 

.88 

3188 

413  , 

0.020 

0.  04146 

-0.00119 

.39 

6211 

241 

.89 

2775 

389 

.022 

4027 

109 

.024 

3918 

100 

0.40 

0.  005970 

-0.  000232 

0.90 

0.  0002386 

-0.  0000363 

.026 

3818 

93 

.41 

5738 

225 

.91 

2021 

341 

.028 

3725 

86 

.42 

5514 

217 

.92 

1680 

316 

.43 

5297 

210 

.93 

1364 

290 

0.030 

3639 

-0.  00081 

.44 

5087 

202 

.94 

1074 

263 

.032 
.034 
.036 
.038 

0.040 

3558 
3482 
3411 
3343 

0.  03279 

76 
71 
68 
64 

-0.00061 

0.45 
.46 
.47 
.48 
.49 

0.  004885 
4690 
4501 
4318 
4140 

-0.  000195 
189 
183 

178 
171 

0.95 
.96 
.97 
.98 
.99 

0.  00008107 
5756 
3710 
2004 
703 

-0.00002351 
2046 
1706 
1301 
703 

.042 

3218 

58 

Olfrt 

0(v"i7Q£iO 

A  fWUfifi 

1.00 

0 

.044 

3160 

55 

.  j(j 

.51 

.  uujyoy 
3803 

~—  u.  lA/vlOO 

160 

.046 

3105 

53 

.52 

3643 

156 

0.950 

O."00008107 

-0.00000494 

.048 

3052 

51 

.53 

3487 

150 

.952 

7613 

482 

.54 

3337 

146 

.954 

7131 

4.70 

0.050 

0.  03001 

-0.  00226 

.956 

6661 

458 

.060 

2775 

191 

0.55 

0.  003191 

-0.  000141 

.958 

6202 

446 

.070 

2584 

164 

.56 

3050 

137 

.080 

2420 

144 

.57 

2913 

133 

0.960 

0.  00005756 

-0.  00000436 

.090 

2276 

128 

.58 

2780 

128 

.962 

5320 

421 

.59 

2652 

125 

.964 

4899 

409 

o-lOO 

.11 

0.  02148 
2032 

-0.00116 
104 

0.60 

0.  002527 

-0.  000120 

.966 
.968 

4490 
4093 

397 
383 

.12 

1928 

96 

.61 

2407 

117 

.13 

1832 

89 

.62 

2290 

113 

0.970 

0.  00003710 

-0.  00000370 

.14 

1743 

82 

.63 

2177 

109 

.972 

3340 

356 

.64 

2068 

106 

.974 

2984 

341 

0.15 
.16 

0.  01661 
1586 

-0.  00075 
71 

0.65 
.66 

0.  001962 
IS59 

-0.  000103 
99 

.976 
.978 

2643 
2316 

327 
312 

.17 
.18 

.19 

1515 
1449 
1387 

66 

62 
59 

.67 
.68 
.69 

1760 
1664 
1571 

96 

93 
'  90 

0.980 
.982 
.984 

0.  00002004 
1708 
1430 

-0.  00000296 
278 
262 

0-20 

0.  01328 

-0.00055 

0.70 

0.  001481 

-0.  000087 

.986 

1168 

242 

.21 

1273 

52 

.71 

1394 

84 

.988 

926 

223 

.22 

1221 

50 

.72 

1310 

81 

.23 

1171 

47 

.73 

1228 

78 

0.990 

0.  00000703 

-0.  00000201 

.24 

1124 

45 

.74 

1150 

76 

.992 

502 

177 

.994 

326 

148 

0.25 

0.  010792 

-0.  000425 

0.75 

0.  0010740 

-0.0000731 

.996 

177 

115 

.26 

10366 

408 

.76 

.  -1001 

704 

.998 

062 

62 

.27 

0.  009958 

388 

.77 

0930 

680 

.28 

9570 

371 

.78 

862 

653 

.29 

9199 

355 

.79 

797 

628 

DESIGN    OF   INDUCTANCE   COILS 
71.  DESIGN  OF  SINGLE-LAYER  COILS 

The  problems  of  design  of  single-layt*<^ils  m£1r  be  broadly 
classified  as  of  two  kinds. 

(i)  Where  it  is  required  to  design  a  coil  which  shall  have  a 
certain  desired  inductance  with  a  given  length  of  wire,  the  choice 
of  dimensions  of  the  winding  and  kind  of  wire  to  be  used  being 
unrestricted  within  rather  broad  limits.  This  class  of  problems 
of  design  includes  a  consideration  of  the  question  as  to  what 


Radio  Instruments  and  Measurements  287 

shape  of  coil  will  give  the  required  inductance  with  the  minimum 
resistance. 

(2)  Given  a  certain  winding  form  or  frame,  what  pitch  of 
winding  and  number  of  turns  will  be  necessary,  if  a  certain 
inductance  is  to  be  obtained. 

In  the  following  treatment  of  the  problem  the  inductance  of 
the  coil  will  be  assumed  as  equal  to  that  of  the  equivalent  cylin- 
drical current  sheet.  This  is  allowable,  since,  in  general,  the 
correction  for  the  cross  section  of  the  wire  will  not  amount  to 
more  than  i  per  cent  of  the  total  inductance,  an  amount  which 
may  be  safely  neglected  in  making  the  design.  The  formulas  to 
be  given  may,  of  course,  be  used  for  making  a  calculation  of  the 
inductance  of  a  given  coil.  Nevertheless,  since  their  practical 
use  is  made  to  depend  upon  the  interpolation  of  numerical  values 
from  a  graph,  for  accurate  calculations  formulas  (153)  and  (155) 
should  be  used. 

The  inductances  of  coils  of  different  size,  but  of  identical  shape, 
and  the  same  number  of  turns,  are  proportional  to  the  ratio  of 
their  linear  dimensions.  Every  formula  for  the  inductance 
should,  accordingly,  be  capable  of  expression  in  terms  of  some 
single  chosen  linear  dimension,  all  the  other  dimensions  occurring 
in  the  formula  in  pairs  in  the  form  of  ratios. 

Two  formulas  are  here  developed,  the  first  applicable  to  the 
solution  of  problems  of  the  first  class,  giving  the  inductance  in 
terms  of  the  total  length  of  wire  /,  the  second  for  problems 
presupposing  a  winding  frame  of  given  dimensions.  Both  show 
the  dependence  of  the  inductance  on  the  shape  of  the  coil. 

Coil  of  Minimum  Resistance.  —  The  fundamental  relations  of  the 
constants  of  a  coil  are 


a2 
L8  =  47T2w2-r-K  cgs  units 

2d 

the  constant  K  being  a  function  of  the  shape  factor  -r-  ,    diameter 

•*-  lengtiigifable  ic^p.  283). 
The  expression  for  the  inductance  may  be  written  as 

__2Traln 
L.--J-K 

and  n  may  be  eliminated  by  substituting  for  it  the  expression 


35601°—  18  -  19 


288 


Circular  of  the  Bureau  of  Standards 


obtained  by  multiplying  together  the  two  expressions  involving 
n  above.     There  results,  then, 


,         /  2O      /       jrr  ., 

Le  =  u    TT  -r~-K      cgs  units 


or 


s  

Z       K       I  Z 

r.  =  — T=T .  / ,_  ?£  ==  -7=  F  microhenries 

yD  loooy      6      v^ 


(194) 


FIG.  207.  —  (j)  Variation  of  F  with  different  ratios  of  coil  diameter  to  length;  (2)  "variations 
of  v  with  ratios  of  diameter  to  length 


To  aid  in  the  use  of  this  formula  the  curve  of  Fig.  207  has  been 

I  - 
-  /TT— 

Y        6 


prepared,  which  enables  the  value  of  F 


IOOO 


to  be  obtained 


2/TT 


for  any  desired  value  of -r  •     The   formula    (194)    and   the   curve 

enable  one  to  obtain  with  very  little  labor  the  approximate  value 
of  the  inductance  which  may  be  obtained  in  a  coil  of  given  shape 
with  given  /  and  D.  On  the  same  figure  is  also  plotted  the  factor 

—  as  a  function  of  —  -  (see  example  below) . 

7T20  b 


Radio  Instruments  and  Measurements 


289 


Coil  Wound  on  Given  Form.  —  To  obtain  the  second  formula,  we 


substitute  for  n  its  value  yy  and 

b2  a2 


2aV  b 


or 


L, 


IT 


IOOO20 


.K"    microhenries 


and,  finally, 


(195) 


l 


:::: 


20, 


FIG.  208. — Variation  of f  and  Iog10[fwith   j- 


To  aid  in  making  calculations  the  curves  of  Fig.  208  have  been 

rT  OOO  2  CL   I 
^.^r ,   W6.w^w«,    VCMUM   ~*  ,    «.^   ^&io/-^&io       ~¥^  IT 

LTT/V    bj 

2  fl 

for  different  values  of  -j--    The  value  of  Iog10  /  is  plotted,  rather 

than  that  of  /,  for  large  values 
polated  with  greater  accuracy. 


than  that  of  /,  for  large  values  of  -r » to  enable  values  to  be  inter- 


290  Circular  of  the  Bureau  of  Standards 

From  formula  (194)  and  Fig.  207  it  is  at  once  evident  that 
with  a  given  length  of  wire,  wound  with  a  given  pitch,  that  coil 
has  the  greatest  inductance,  which  has  such  a  shape  that  the 

diameter 
ratio  -;  —    .     =2.46  approximately.    Or,  to  obtain  a  coil  of  a 

certain  desired  inductance,  with  a  minimum  resistance,  this 
relation  should  be  realized.  However,  although  the  inductance 
diminishes  rather  rapidly  for  longer  coils  than  this,  changes  in 
the  direction  of  making  the  coil  shorter  relative  to  the  diameter 
are  not  important  over  rather  wide  limits.  Naturally,  other 
considerations  may  modify  the  design  appreciably.  These  other 
considerations  include  the  distributed  capacity  of  the  coil  and 
the  variation  of  resistance  with  frequency. 

Example.  —  Given  the  pitch  of  winding,  the  shape  of  the  coil  (  T~  )' 

and  the  inductance,  to  determine  the  length  of  wire  necessary,  the 
dimensions  of  the  coil  and  the  number  of  turns. 

Assuming  D  =  0.2  cm,  -r-  =  2.6,  L8  =  1000  microhenries, 


By  formula  (194)  ,  l\  =  -  i  (the  value  of  F  =  0.001322  being 

0.001322 

log  1000  =  3.  taken   from   the   curve   of   Fig.    207)    or 

Klog  0.2    =1.65052     2  =  4850  cm.     The  number  of  turns  may 

2  6^052     ke  obtained  immediately  from  the  relation 

log  F       =  3.12123     n  =    fi  .    /  A.  =  „    /J  and  the  graph  of  v. 

\D\  2ira        \  D 
3/2log/          =5.52929 

=1.84310 


log  I          =3.68619 

Here  n  =  ^r~  $°  (0.350)  =54.5  turns,  and  b=nD  =  io.<)  cm,  while 
Y   0.2 

20  =  2.6  X  10.9  =  28.3  cm. 

If  the  pitch  of  the  winding  had  been  assumed  greater,  or  a  coil  of 
much  larger  inductance  were  required,  the  design  of  the  coil  would 
call  for  larger  dimensions,  and  cases  may  arise  where  the  design 
may  prove  unsatisfactory,  because  the  coil  would  be  too  large. 
The  effect  of  changing  the  length  and  pitch,  the  shape  being  taken 

/I 
constant,  may  be  seen  from  (194),  which  shows  that  L»oc  -T=,  so 

that  a  given  fractional  increase  in  the  length  of  the  wire  is  more 


Radio  Instruments  and  Measurements  291 

effective  in  increasing  the  inductance  than  the  same  fractional 
decrease  in  the  pitch.  The  number  of  turns  depends  on  -  /—  the 

shape  of  the  coil  being  kept  the  same. 

Example. — Formula  (194)  will  also  enable  the  question  to  be 
answered  as  to  what  pitch  must  be  used  if  a  given  length  of  wire 
is  to  be  wound  with  a  certain  shape  of  coil  to  give  a  desired 
inductance.  If  the  pitch  comes  out  smaller  than  the  diameter  of 
the  proposed  wire,  the  assumed  length  of  wire  must  be  increased. 

Suppose  that  an  inductance  of  10  ooo  microhenries  is  desired 

2a 
with  50  meters  of  wire,  the  value  of  -j-  being  taken  as  2.6,  as  before. 

Then 

l\  _     (5000)*  0.001322 
>T-F  — -  — - — »  or  /}  =0.00218  cm, 

L8  10  ooo 

which  is  manifestly  impracticably  small. 

The  maximum  inductance  attainable  with  the  given  length  of 
wire  could  be  found  by  solving  (194)  for  L  with  the  smallest 
practicable  pitch  substituted  for  D,  that  value  being  used  for  F, 
which  corresponds  to  the  assumed  ratio  of  diameter  to  length. 

Example. — Suppose  we  have  a  winding  form  of  given  diameter 
2a  =  io  cm,  how  many  turns  of  wire  will  have  to  be  used  for  an 
inductance  of  looonh  if  the  winding  pitch  is  taken  as  0.2,  and 
what  will  be  the  axial  length  of  the  winding  ? 

From  (196) 

1000 

/  =  —  -  =25  or  Iog10/  =  i.398 

1000X0.04 

From  Fig.  208  this  corresponds  to  a  value  of  -r=  0.225,  or  6 

must  be  45  cm,  and  the  number  of  turns  n  =  jj  =  —  =  225.    Such  a 

coil  would  be  too  long  to  be  convenient.  A  smaller  pitch  should 
be  used. 

Example. — Suppose  we  have  given  the  same  winding  form,  and 
we  wish  to  find  what  pitch  is  necessary  for  an  inductance  of 
loooph,  in  order  that  the  length  of  the  coil  shall  not  be  greater 
than  the  diameter. 

For 

2<X 

y=  I,/ =I48   (Fig.  208) 

and  by  (196) 

^.,     (2a}?-  looo 

D9  =  — — —  = 77  or  D  =  0.082 

Ltf       1000X148 


292  Circular  of  the  Bureau  of  Standards 

This  is  a  pretty  close  winding,  showing  that  the  winding  form  has 
rather  too  small  a  diameter  for  a  coil  of  this  inductance. 

Example.  —  To  find  the  diameter  of  a  winding  form  to  give  an 


inductance  of  looonh,  with  a  shape  ratio  -r  =  2.6,  the  pitch  being 

chosen  as  0.2  cnr. 

From  (196)  we  have  (2a)3  =  LBD2. 

2a 
The  value  of  /  for  y  =  2.6  is  (from  Fig.  208)  given  by  Iog10  /  = 

2.75   or  7  =  565   approximately.     Therefore    (2a)3  =  ioooXo.O4X 
565,  or  20  =  28.2  cm,  which  will  give  6  =  10.85,  ^  =  54.2. 

2Q, 

If,  instead,  the  shape  is  assumed  to  be  given  by  -r-  =  i  ,  then 


log  /  =  2.iy  or  /  = 

(20)  3  =  looo  X  0.04  X  148,  or  2a  =  i8.i  cm  =  6,  and  n  =  9O.5. 

The  values  of  /  taken  from  Fig.  208  are  not  so  precise  as  could 
be  calculated  from  the  equation  (195),  but  the  accuracy  should 
suffice  for  this  kind  of  work. 

72.  DESIGN  OF  MULTIPLE-LAYER  COILS 

For  purposes  of  design  we  may  neglect  the  correction  for  cross 
section  of  the  wire,  formula  (159),  and  operate  on  formulas  (157) 
and  (158)  alone. 

Two  forms  of  equation  have  been  found  useful,  the  first  involving 
the  length  of  wire  in  the  coil  and  the  second  the  mean  radius  of 
the  coil. 

Suppose  that  the  length  of  the  winding  I,  the  distance  between 

7 

the  centers  of  adjacent  wires  D,  shape  of  cross  section  -,  and  the 

C- 

shape  ratio  of  the  coil  -»  are  given.     We  obtain  an  expression  for 

a 

n  by  multiplying  together  the  fundamental  equations, 
be      b/c  I2 


b/c\ 
-i7^) 
c\D/ 


which  involves  ratios  of  known  quantities  only. 


Radio  Instruments  and  Measurements 


293 


In  equation  (158)  the  factor  47ran2  =  2/n,  and  if  the  value  of  n 
just  found,  be  introduced,  we  have  finally  for  c>b 


(198) 


i6a: 


}V 


and  for  b  >  c 


L-+  -. 


2 


(r99) 


FIG.  209.  —  Values  of  (G)for  given  -values  of  c_an(lb_ 

a         c 

Both  of  these  equations  may  be  written  in  the  form 

/! 
L  =  =G  microhenries 


in  which  G  is  a  factor  whose  value  for  given  values  of  -  and  -  may 

CL  C 

be  taken  from  the  curves  of  Fig.  209. 


294 


Circular  of  the  Bureau  of  Standards 


When  /  is  known 


c    D* 


/      /     M 

(c/a)2 


(201) 


From  these  curves  one  can  see  that,  for  a  square  cross  section, 
b/c  =  i,  the  inductance  of  a  given  length  of  wire  is  a  maximum  for 

C  2 

a  value  of  -  equal  to  about  —     Investigation  shows  that   this 

O  , 

point  is,  more  exactly,  c/a  =  0.662;  that  is,  for  a  mean  diameter  of 
coil  =  3.02  times  the  side  of  the  cross  section.  Further,  for  a  given 
resistance  and  shape  of  coil,  the  square  cross  section  gives  a  greater 
inductance  than  any  other  form. 


tit 


0)6 


FIG.  sic.— Values  of  (g)for  given  -values  of  —  and  — 

a          c 

The  second  design  formula  supposes  that  the  dimensions  a,  c, 
and  -  of  the  winding  form  are  given,  together  with  the  pitch  of  the 

C 

winding.     The  expressions  (157)  and  (158)  for  the  inductance  may 
then  be  written 


L  =0.01257  a 


=0.01257  an  9 


microhenries 


(202) 
(203) 


Radio  Instruments  and  Measurements  295 

The   curves   of    Fig.    210,  which  give  g  for  different  values    of 
-  and  -  allow  of  interpolation  of  the  proper  value  in  any  given 

CL  C 

case. 

Example. — Suppose  we  have  a  wire  of  such  a  size  that  it  may  be 
wound  20  turns  to  the  centimeter,  and  we  wish  to  design  a  coil  to 
have  an  inductance  of  10  millihenries,  to  have  a  square  cross 
section  and  such  a  mean  radius  as  to  obtain  the  desired  inductance 
with  the  smallest  resistance  (smallest  length  of  the  wire). 

The  latter  condition  requires  that  -=0.662.     The  given  quan- 

CL 

tities  are  .0=0.05  cm,  b/c  —  i.     From  Fig.  209  we  find  that  G  = 

0.000606,  so  that  (200)   becomes   10000  =  7 rj  0.000606,  from 

which  /  =  6458  cm  or  64.58  meters  of  wire. 

2/3  log  D  =  1.13265         From  the  fundamental  equation     (201) 
io7 


5/3  log  /  =  6.35018 

1/3  log/  =^27004  "' 

2  log  /  =  7.  62O22 

log/  =  3.8101  1  =1.80 

and  thence  6=c  =  o.662  X  1.80  =  1.19,  and  n  =  j^  —  —   -^—  =  570. 

D2     0.0025 

This  coil  is  rather  too  small  to  allow  of  its  dimensions  being 
accurately  measured.  § 

If  wire  of  double  the  pitch  is  used,  the  design  works  out  with 
the  following  results 

/  =  85.  22  meters          c  =  6  =  2.o8 
^  =  432  0  =  3.18 

which  is  more  suitable. 

Example.  —  We  have  a  form  whose  dimensions  are  2<z  =  10,  £  =  3, 
b  =  2.4,  wound  with  wire  of  such  a  size  that  there  are  10  turns  per 
cm;  that  is,  D=o.i.  What  is  the  inductance  obtained  and  what 
length  of  wire  is  used  ? 

be      3X2.4 

J 


— 


T-»O  — 

D3      o.oi 


296  Circular  of  the  Bureau  of  Standards 

From  Fig.  210  the  interpolated  value  of  g  for  -  =  o.8,  c/o  =  o.6,  is 

O 

1.54  (calculated  directly  from  (158)  =  1.552).     Accordingly, 

L  =0.01  257x5X72?  Xi.  54  =  50  1  60  ph. 

=  50.  1  6  millihenries. 


The  length  of  wire  isl  =  2Tran  =  ioir  720.=  22  600  cm 

=  226  meters. 

Example.  —  The  same  formula  might  be  used  to  answer  the  ques- 
tion, How  many  turns  would  have  to  be  wound  (completely  filling 
this  cross  section)  in  order  to  obtain  a  desired  inductance,  say  20 
millihenries.  From  (203), 

L  20  ooo 

ft*  =  -  =  7  -  v  -  7  -  r  =  2OO   SOO 

0.01257  ag     (0.01257)  5  (i.54) 
or  n  would  be  454,  which  would  mean  that 

be      7.20 
2  — 


454      454 


=0.0158 


or  .0=0.126,  so  that  the  wire  would  have  to  wind  about  8  turns 
to  the  centimeter. 

The  skin  effect  and  capacity  between  the  layers  of  the  wire  are 
larger  in  this  kind  of  coil  than  in  the  other  forms  previously  con- 
sidered. A  multiple  layer  coil  is  therefore  to  be  regarded  as  unde- 
sirable in  radio  work,  and  if  it  be  used  the  cross  section  should  be 
made  small  relative  to  the  mean  radius. 

73.  DESIGN  OF  FLAT  SPIRALS 

The  design  of  a  flat  spiral  differs  from  that  of  a  multiple  layer 
coil  in  that  the  actual  width  b  of  the  tape  used  (not  b/c)  is  sup- 
posed to  be  a  given  quantity. 

The  fundamental  equations  are 


c  I 

^  and  n  =  --  > 
D  2ira 


which,  on  multiplication,  give 


£.1 

a  'D  (204) 


Radio  Instruments  and  Measurements  297 

and  this  introduced  into  the  expression  47rcm2  =  2/n  gives  finally 

g<  8~logf 


(205) 


=  — pL//  microhenries. 


FIG.  211. —  Value  of  (H)  for  given  values  of  —and— 

The  factor  H,  which  may  be  determined  from  the  curves  of  Fig. 
211  is  a  function  of  c/a  and  b/c.  The  latter  quantity  may  be 
expressed  in  terms  of  the  known  quantities  by  the  equation 


-VI 


(206) 


Accordingly,  the  curves  are  plotted  with  H  as  ordinates,  c/a  as 

,  ,      /27r 
abscissas,  and  6  -%//n  as  parameter. 

An  important  deduction  which  may  be  made  from  the  curves  is 
that  for  the  maximum  inductance  with  a  given  length  of  tape  the 
ratio  c/a  should  be  about  ^,  which  means  that  the  opening  of  the 
spiral  should  have  a  radius  nearly  as  great  as  the  dimension  across 


298  Circular  of  the  Bureau  of  Standards 

the  turns  of  the  spiral.  This  point  in  design  is  in  agreement  with 
the  practical  observation  that  turns  in  the  center  of  the  spiral 
add  a  disproportionate  amount  to  the  high-frequency  resistance  of 
the  spiral. 

Example.  —  Find  the  length  of  tape  0.6  cm  wide,  wound  with  a 
pitch  of  0.6  cm,  to  give  an  inductance  of  200  ph,  assuming  such 
proportions  that  cja  =  i  .  Work  out  the  design. 


Since  /  is  not  known,  the   parameter  6  -%T     is  not  known. 

Assume  a  value  of  o.i  for  the  latter.     Then  for  the  value  c/a=  i 
the  curve  (Fig.  211)  gives  H  =  0.00123. 

Thence  /T  =  -          -  or  £  =  3287  cm.     With  this  value  of  /,  the 


parameter  is  0.6  -t/-    -  or  0.0339,  to  which  the  value  H  =  0.00128 


corresponds  (with  -  ==  i).     Repeating  the  calculation  of  /  with  this 

CL 

value  of  H,  we  find  £  =  3370  cm  as  a  second  approximation.  The 
next  approximation  gives  a  parameter  of  0.0335  and  the  values  of 
H  and  /  are  sensibly  unchanged. 

Using  this  parameter  in  (206)  ,  -  =0.0335  orc  =  -       —  =17.  9  and 

"     \)OO 

the  value  of  a  =  1  7  .9  likewise.     The  number  of  turns  will  be  n  =      ' 

0.6 

=  about  30. 

Example.  —  We  have  17.50  meters  of  tape  i  cm  wide,  which  we 
wind  with  a  pitch  of  0.5  cm,  to  such  a  shape  that  c/a=o.8. 

Here  D  =  0.5,  /  =  1  750  cm,  b  =  i  .    The  parameter  is  */:j-     =  0.0847, 

V  °75 
to  which,  for  c/a  =  o.8,  H  =  0.001248  corresponds. 


( 
L  =  —  ,^—  o.ooi  248  =  1  29.2 

Vo-5 

b     0.0847 

-  =          -  =  0.0947,  by  equation  (206) 

£         -\O.o 

T 

=  10.56  cm. 


0.0947 

10.56 
°  =  ^8-:='3-2 

and  the  number  of  turns,  n  —  — —  =  21  nearly. 

0.5  * 


Radio  Instruments  and  Measurements  299 

Example. — The  problem  may  arise  as  to  how  closely  the  tape 

in  the  preceding  case  would  have  to  be  wound,  still  keeping  -  =0.8, 

a 

to  obtain  an  inductance  of  200  ph. 

Changing  the  pitch  D  will  change  the  parameter  of  the  curves, 
and  hence  H.  The  changes  in  the  latter  will  not  be  important, 
for  small  changes  in  D,  so  that  to  a  first  approximation  the  induc- 
tance will  change  inversely  as 

Therefore 


[D      129.2 

*/  —  =-    — t  or  D=  0.2086  cm. 
Yo.5       200 


Calculating  the  parameter  with  this  value  we  find  0.1312, 
and  thence  H= 0.00121 6,  so  that  the  second  approximation  is 

/  \    3 

^JD  = —         (0.001216),  and  D=  0.1981,  and  another  approxima- 
200 

tion  is  0.197,  the  parameter  being  0.1346.     The  dimensions  are 
found  from 

6     0.1346  i 

-  =  — ^^  =  0.1505         c— —  =  6.64 

c       Vo.8  0.1505 

c  6.649 

a  =  —^  =  8.30  n  =  -    -  =  34  nearly. 

0.8  0.197 

HIGH-FREQUENCY  RESISTANCE 
74.  RESISTANCE  OF  SIMPLE  CONDUCTORS 

Two  principal  causes  act  to  increase  the  resistance  of  a  cir- 
cuit carrying  a  current  of  high  frequency,  above  the  value  of 
its  resistance  with  direct  current,  viz,  the  so-called  skin  effect 
and  the  capacity  between  the  conductors.  This  section  deals  ex- 
clusively with  the  skin  effect  or  change  of  resistance  caused  by 
change  of  current  distribution  within  the  conductor.  (See  sec.  3.) 

Unfortunately,  formulas  for  the  skin  effect  are  available  only 
for  the  most  simple  circuits;  and  for  other  very  common  cases  in 
practice  only  qualitative  indications  of  the  magnitude  of  the 
increase  in  resistance  can  be  given. 

In  what  follows 

R  =the  resistance  at  frequency  / 

JR0  =  the  resistance  with  direct  current  or  very  low  frequency 
alternating  current. 


300  Circular  of  the  Bureau  of  Standards 

The  quantity  of  greatest  practical  interest  is  not  R,  but  the 

D 

resistance  ratio  •==-  •     Given  this  ratio  for  the  desired  frequency  and 

the  easily  measured  direct-current  resistance,  the  high-frequency 
resistance  follows  at  once. 

The  skin  effect  in  a  conductor  always  depends,  in  addition  to 

the  thickness  of  the  conductor,  on  the  parameter  -v       »  in  which 

H  =  permeability  of  the  material,  /  =  frequency  of  the  current, 
p  =  the  volume  resistivity  in  microhm-cms,  so  that  as  far  as  skin 
effect  is  concerned,  a  thick  wire  at  low  frequencies  may  show  as 
great  a  skin  effect  as  a  thin  one  at  much  higher  frequency. 

The  skin  effect  is  greater  in  good  conductors  than  in  wires  of 
high  resistivity,  and  conductors  of  magnetic  material  show  an 
exaggerated  increase  of  resistance  with  frequency. 

Cylindrical  Straight  Wires. — For  this  case  accurate  values  of 
the  resistance  ratio  are  given  by  the  formula  and  tables  here 
given. 

If  d  is  the  diameter  of  the  cross  section  of  the  wire  in  cm,  the 
quantity 


must  be  calculated  (or,  in  the  case  of  copper,  obtained  for  the 

desired  frequency  from  Table  1 9 ,  p .  3 1 1  and  formula  ( 209) ) .     Know- 

r> 
ing  the  value  of  x,  the  value  of  ^r  may  be  taken  at  once  from  Table 

K0 

r> 

17,  page  309,  which  gives  the  value  of  •_-  directly  for  a  wide  range 

/Co 

of  values  of  x. 

Table  1 9  gives  values  of 

ac=o.io7i-y/r  (208) 

for  a  copper  wire  at  20°  C,  o.i  cm  in  diameter,  and  at  various 
frequencies.  The  value  of  x  for  a  copper  wire  of  diameter  d  in 
cm  is 

xc  =  iodac  (209) 

For  a  material  of  resistivity  p  and  permeability  ju,  the  parameter 
x  may  also  be  simply  obtained  from  the  value  which  holds  for  a 
copper  wire  of  the  same  diameter,  by  multiplying  the  latter  value 


Radio  Instruments  and  Measurements  301 

The  range  of  Table  19  may  be  considerably  extended  by  remem- 

bering that  a  is  proportional  to  V/  or  \Ar'  wnere  X  is  the  wave 

length. 

Table  18,  page  310,  will  be  found  useful,  when  it  is  desired  to 
determine  what  is  the  largest  diameter  of  wire  of  a  given  mate- 
rial, which  has  a  resistance  ratio  of  not  more  than  i  per  cent 
greater  than  unity.  These  values  are,  of  course,  based  on  certain 
assumed  values  of  resistivity;  temperature  changes  and  differ- 
ences of  chemical  composition  will  slightly  alter  the  values.  In 
the  case  of  iron  wires  /*  is  the  effective  permeability  over  the 
cycle.  This  will,  in  general,  be  impossible  to  estimate  closely. 
The  values  given  show  plainly  how  important  is  the  skin  effect 
in  iron  wires. 

For  a  resistance  ratio  only  one-tenth  per  cent  greater  than 
unity  the  values  in  Table  18  should  be  multiplied  by  0.55,  and 
for  a  10  per  cent  increase  of  the  high-frequency  resistance  the 
diameters  given  in  the  table  must  be  multiplied  by  1.78. 

The  formulas  above  given  apply  only  to  wires  which  are  too  far 
away  from  others  to  be  affected  by  the  latter.  For  wires  near 
together,  as,  for  example,  in  the  case  of  parallel  wires  forming  a 
return  circuit,  the  mutual  effect  of  one  wire  on  the  other  always 

TO 

increases  the  ratio  -p-     No  formula  for  calculating  this  effect  is 

-fVo 

available,  but  it  is  only  for  wires  nearly  in  contact  that  it  is  impor- 
tant. At  distances  of  10  to  20  cm  the  mutual  effect  is  entirely 
negligible. 

Tubular  Conductors.  —  The  resistance  ratio  of  tubular  conduc- 
tors in  which  the  thickness  of  the  walls  of  the  tube  is  small  in  com- 
parison with  the  mean  diameter  of  the  tube,  may  be  calculated 
by  the  theoretical  formula  for  an  infinite  plane  of  twice  the  thick- 
ness of  the  walls  of  the  tube. 

The  value  of  the  resistance  ratio  for  this  case  may  be  obtained 
directly  from  Table  20,  page  311,  in  terms  of  the  quantity 


where 

T  =  the  thickness  of  the  walls  of  the  tube  in  cm 
*  x  =  the  parameter  defined  in  formula  (207)  . 

For  copper  tubes  the  parameter  $c   may  be  obtained  very 
simply  from  the  values  of  ac  in  Table  19,  page  31  1  ,  and  the  relation 


302  Circular  of  the  Bureau  of  Standards 

For  values  of  /3  greater  than  4  no  table  is  necessary,  since  we 
have  simply,  with  an  accuracy  always  greater  than  one-tenth  of 
i  per  cent, 


Sufficient  experimental  evidence  is  not  available  to  indicate  an 
accurate  method  of  procedure  in  the  case  of  tubing  where  the 
ratio  of  diameter  to  wall  thickness  is  not  large.  Measurements 
with  tubing  in  which  this  ratio  is  as  small  as  two  or  three  indicate 

TJ 

that  approximate  values  of  -p-  for  this  case  may  be  calculated  by 

KO 

using  for  T,  in  the  calculation  of  the  parameter  /3,  a  value  equal  to 
two-thirds  of  the  actual  thickness  of  the  walls  of  the  tube. 

Tubing  which  is  very  thin  in  comparison  with  its  radius  has, 
for  the  same  cross  section,  a  smaller  high-frequency  resistance 
than  any  other  single  conductor.  For  this  reason  galvanized- 
iron  pipe  is  a  good  form  of  conductor  for  some  radio  work,  the 
current  all  flowing  in  the  thin  layer  of  zinc.  A  conductor  of 
smaller  resistance  than  a  tube  of  a  certain  cross  section  is  obtained 
by  the  use  of  very  fine  strands  separated  widely  from  one  another; 
there  are  practical  difficulties,  however,  in  making  the  separation 
great  enough. 

In  a  return  circuit  of  tubular  conductors  the  distance  between 
the  conductors  should  be  kept  as  great  as  i  o  or  20  cm.  For  tubular 
conductors  nearly  in  contact  the  resistance  ratio  may  be  double 
that  for  a  spacing  of  a  few  centimeters. 

I  x. 


"i 
* +• 

-4- 
T- T 


FIG.  212. — Cross  section  of  strip  conductors  forming  a  return  circuit  with  narrow  surfaces 

in  the  same  plane 


Strip  Conductors. — If  two  strips  form  together  a  return  circuit 
and  they  are  so  placed  that  there  is  only  a  small  thickness  of 
dielectric  between  the  wider  face  of  one  and  the  same  face  of  the 
other  (Fig.  212),  the  resistance  ratio  may  be  calculated  by  formula 
(210) ,  using  for  r  the  actual  thickness  of  the  strip. 


Radio  Instruments  and  Measurements 


303 


As  the  thickness  of  the  insulating  space  between  the  plates  is 
increased,  the  accuracy  of  the  formula  decreases,  but  the  error 
does  not  amount  to  more  than  a  few  per  cent  for  values  of  this 
thickness  as  great  as  several  centimeters. 


W 


-w- 


* — 7- 

<L  * 


FlG.  213. — Cross  section  of  strip  conductors  forming  a  return  circuit  with  wide  surfaces  in 

the  same  plane 

For  a  return  circuit  of  strips  placed  with  their  wider  faces  in 
the  same  plane  (Fig.  213),  no  formula  is  available.  This  is  an 
unfavorable  arrangement.  As  the  distance  t  is  reduced  below  a 

few  centimeters  the  ratio  ^-  increases  rapidly  and  with  the  strips 

very  close  together  may  be  as  great  as  twice  the  value  for  the 
arrangement  of  Fig.  212. 

For  single  strips — that  is,  for  return  circuits  in  which  the 
distance  between  the  conductors  is  so  great  that  there  is  no 
appreciable  mutual  effect  between  the  conductors — formula  (210) 
is  inapplicable  owing  to  "edge  effect" — the  effect  of  the  magnetic 
field  produced  by  the  current  in  the  center  of  the  strip  upon  the 
outer  portions  of  the  cross  section. 

r> 

Thus  the  resistance  ratio  -5-  is  greater  in  a  wide  strip  than  in  a 

ZV0 

narrow  one  of  the  same  thickness,  and  in  every  case  the  resistance 
ratio  is  greater  than  for  the  two  juxtaposed  strips  of  Fig.  212. 

R  ^ 

For  -p-  between   i   and   1.5,  the  increase  over  formula   (210)   is 


E> 


usually  not  greater  than  10  per  cent. 

Strips  of  square,  or  nearly  square,  cross  section  have  values  of 

not  very  different  from  those  which  hold  for  round  conductors  of 
the  same  area  of  cross  section,  the  values  being  greater  for  the 
square  strip  than  for  the  round  conductor  whose  diameter  is  equal 
to  the  side  of  the  square. 

Simple  Circuits  of  Round  or  Rectangular  Wire. — The  ratio  of  the 
resistance  at  high  frequencies  to  that  with  direct  current  may  be 
accurately  obtained  from  Table  1 7,  page  309,  for  circles  or  rectangles 
of  round  wire  and  in  fact  for  any  circuit  of  which  the  length  is 

35601°— 18 20 


304  Circular  of  the  Bureau  of  Standards 

great  compared  with  the  thickness  of  the  wire,  provided  no  con- 
siderable portions  of  the  circuit  are  placed  close  together.  In  the 
latter  case,  the  resistance  ratio  is  somewhat  increased  beyond  the 
value  calculated  by  the  previous  method  and  by  an  amount  which 
can  not  be  calculated. 

The  resistance  ratio  for  a  circuit  of  wire  of  rectangular  section 
may  be  treated  by  the  same  method  as  for  a  single  strip.  If  por- 
tions of  the  circuit  are  in  close  proximity,  the  precautions  men- 
tioned for  two  strips  near  together  (p.  303)  should  be  borne  in  mind. 

75.  RESISTANCE  OF  COILS 

Single-Layer  Coil;  Wire  of  Rectangular  Cross  Section. — The 
only  case  for  which  an  exact  formula  is  available  is  that  of  a 
single-layer  winding  of  wire  of  rectangular  cross  section  with  an 
insulation  of  negligible  thickness  between  the  turns,  the  length  of 
the  winding  being  assumed  to  be  very  great  compared  with  the 
mean  radius,  and  the  latter  being  assumed  very  great  compared 
with  the  thickness  of  the  wire. 

If     R  =  the  resistance  at  high  frequency 
R0  =  the  resistance  to  direct  current 
T  =  the  radial  thickness  of  the  wire 
b  =  the  axial  thickness  of  the  wire 
p  =  the  volume  resistivity  of  the  wire  in  microhm-cm 
pc  =  the  volume  resistivity  of  copper 
H  =  the  permeability  of  the  wire 
D  =the  pitch  of  the  winding, 

E> 

then  i^-  may  be  obtained  directly  from  Table  20,  page  311,  having 

KO 

r-  •     /M/ 

calculated  first  the  quantity  /3  =  lory  2  a,  in  which  a  =  0.1985  -»/ - 

Values  of  ac  for  copper  are  given  in  Table  19,  page  311,  and  the 
value  of  a  for  any  other  material  is  obtained  from  ac  by  the  relation 

a  =  ac  A  p.  —  •     For  values  of  /3  greater  than  are  included  in  Table 
V     P 

20  we  have  simply  jr~  =  $. 

KO 

In  practice  the  ideal  conditions  presupposed  above  will  not  be 
realized.  To  reduce  the  value  calculated  for  the  idealized  wind- 
ing corrections  need  to  be  applied:  (i)  For  the  spacing  of  the 
wire,  (2)  for  the  round  cross  stection  of  the  wire,  (3)  for  the  curv- 
ature of  the  wire,  (4)  for  the  finite  length  of  the  coil. 


Radio  Instruments  and  Measurements  305 

Correction  for  Pitch  of  the  Winding.  —  To  take  into  account  the 
fact  that  the  pitch  of  the  winding  is  not  in  general  equal  to  the 
axial  breadth  of  the  wire  an  approximation  is  obtained  if  for  j8 
the  argument 

is  substituted. 

D 
For  values  of  D  greater  than  about  36  the  values  of  TT  thus 

•*Vo 

obtained  are  too  small. 

Correction  for  the  Round  Cross  Section  of  the  Wire.  —  For  coils 
of  round  wire  only  empirical  expressions  are  known,  and  more 
experimental  work  is  desirable. 

To  obtain  an  accuracy  of  perhaps  10  per  cent  in  the  resistance 
ratio  the  following  procedure  may  be  used: 

Calculate  first  by  (210)  and  Table  20,  page  311,  the  resistance 

r>/ 

ratio  p-/'  supposing  the  coil  to  be  wound  with  wire  of  square 

-IVO 

cross  section  of  the  same  thickness  as  the  actual  diameter,  taking 
into  account  the  correction  for  the  pitch  of  the  winding.     Then 

r> 

the  resistance  ratio  -^-  for  a  winding  of  round  wire  will  be  found 

•ft-o 

by  the  relation 

-R'-R<,'-  /   x 

(2,2) 


Effect  of  Thickness  of  the  Wire.  —  Although  formula  (210)  holds 
only  for  a  coil  whose  diameter  is  very  great  in  comparison  with 
the  thickness  of  the  wire,  the  error  resulting  from  non-fulfillment 
of  this  condition  will,  in  practical  cases,  be  small  compared 
with  the  other  corrections  and  may  be  neglected. 

Correction  for  Finite  Length  of  the  Coil.  —  For  short  coils  the 
resistance  ratio  is  greater  than  for  long  coils  of  the  same  wire, 
pitch,  and  radius,  due  to  the  appreciable  strength  of  the  magnetic 
field  close  to  the  wires  on  the  outside  of  the  coil. 

No  formulas  are  available  for  calculating  this  effect,  but 
experiment  seems  to  show  that  for  short  coils  of  thick  wire  at 
radio  frequencies  the  resistance  ratio  may  be  expressed  by 

R       A      B  ,       v 


in  which  the  first  term  represents  the  value  as  calculated  by  the 
formulas  of  the  preceding  section  for  long  coils,  while  the  con- 


306  Circular  of  the  Bureau  of  Standards 

stant  of  the  second  term  has  to  be  obtained  by  experiment.  At 
long  wave  lengths  the  first  term  will  predominate,  but  at  very 
short  wave  lengths  the  second  term  may  be  equal  or  even  larger 
than  the  first. 

For  round  copper  wires  we  may  obtain  the  constant  A  by  the 
relation  A  =  1 5  500  dR0. 

Multiple-Layer  Coils. — For  this  case  no  accurate  formulas  have 
been  derived.  Experiment  shows  that  the  resistance  ratio  is 
much  greater  for  a  multiple-layer  coil  than  for  a  single-layer  coil 
of  the  same  wire.  Furthermore,  the  capacity  of  such  a  coil  has, 
as  already  pointed  out,  a  large  effect  on  the  resistance  of  the 
coil.  Consequently,  it  is  usually  impossible  to  calculate  even 
an  approximate  value  for  the  change  of  resistance  with  frequency. 
At  very  high  frequencies  losses  in  the  dielectric  between  the  wires 
may  cause  an  appreciable  increase  in  the  effective  resistance  of  the 
coil.  This  effect  is  proportional  to  /3. 

76.  STRANDED  WIRE 

The  use  of  conductors  consisting  of  a  number  of  fine  wires  to 
reduce  the  skin  effect  is  common.  The  resistance  ratio  for  a 
stranded  conductor  is,  however,  always  considerably  larger  than 
the  value  calculated  by  Table  19,  page  311,  and  Table  17, 
page  309,  for  a  single  one  of  the  strands.  Only  when  the  strands 
are  at  impracticably  large  distances  from  one  another  is  this 
condition  even  approximately  realized. 

Formulas  have  been  proposed  for  calculating  the  resistance 
ratio  of  stranded  conductors,35  but  although  they  enable  quali- 
tatively correct  conclusions  to  be  drawn  as  to  the  effect  of  chang- 
ing the  frequency  and  some  of  the  other  variables,  they  do  not 
give  numerical  values  which  agree  at  all  closely  with  experiment. 
The  cause  for  this  lies,  probably,  to  a  large  extent  in  the  impor- 
tance of  small  changes  in  the  arrangement  of  the  strands.  The 
following  general  statements  will  serve  as  a  rough  guide  as  to 
what  may  be  expected  for  the  order  of  magnitude  of  the  resist- 
ance ratio  as  an  aid  in  design,  but  when  a  precise  knowledge  of 
the  resistance  ratio  is  required  in  any  given  case  it  should  be 
measured.  (See  methods  given  in  sections  46  to  50.) 

Bare  Strands  in  Contact. — The  resistance  ratio  of  n  strands  of 
bare  wire  placed  parallel  and  making  contact  with  one  another  is 
found  by  experiment  to  be  the  same  as  for  a  round  solid  wire 

'"See  references  112  to  123  of  the  Bibliography. 


Radio  Instruments  and  Measurements  307 

which  has  the  same  area  of  cross  section  as  the  sum  of  the  cross- 
sectional  areas  of  the  strands;  that  is,  n  times  the  cross  section 
of  a  single  strand.  This  will  be  essentially  the  case  in  conductors 
that  are  in  contact  and  are  poorly  insulated,  except  that  at  high 
frequencies  the  additional  loss  of  energy  due  to  heating  of  the 
imperfect  contacts  by  the  passage  of  the  current  from  one  strand 
to  another  may  raise  the  resistance  still  higher. 

Insulated  Strands. — As  the  distance  between  the  strands  is 
increased,  the  resistance  ratio  falls,  rapidly  at  first,  and  then 
more  slowly  toward  the  limit  which  holds  for  a  single  isolated 
strand.  A  very  moderate  thickness  of  insulation  between  the 
strands  will  quite  materially  reduce  the  resistance  ratio,  provided 
conduction  in  the  dielectric  is  negligible. 

Spiraling  or  twisting  the  strands  has  the  effect  of  increasing  the 
resistance  ratio  slightly,  the  distance  between  the  strands  being 
unchanged. 

Transposition  of  the  strands  so  that  each  takes  up  successively 
all  possible  positions  in  the  cross  section — as  for  example,  by 
thorough  braiding — reduces  the  resistance  ratio  but  not  as  low 
as  the  value  for  a  single  strand. 

Twisting  together  conductors,  each  of  which  is  made  up  of  a 
number  of  strands  twisted  together,  the  resulting  composite  con- 
ductor being  twisted  together  with  other  similar  composite  con- 
ductors, etc.,  is  a  common  method  for  transposing  the  strands 
in  the  cross  section.  Such  conductors  do  not  have  a  resistance 
ratio  very  much  different  from  a  simple  bundle  of  well-insulated 
strands. 

The  most  efficient  method  of  transposition  is  to  combine  the 
strands  in  a  hollow  tube  of  basket  weave.  Such  a  conductor  is 
naturally  more  costly  than  other  forms  of  stranded  conductor. 

Effect  of  Number  of  Strands. — With  respect  to  the  choice  of  the 
number  of  strands,  experiment  shows  that  the  absolute  rise  of 
the  resistance  in  ohms  depends  on  the  diameter  of  a  single  strand, 
but  is  independent  of  the  number  of  strands.  Since,  however, 
the  direct-current  resistance  of  the  conductor  is  smaller  the  greater 
the  number  of  the  strands,  the  resistance  ratio  is  greater  the 
greater  the  number  of  strands.  Reducing  the  diameter  of  the 
strands  reduces  the  resistance  ratio,  the  number  of  strands  remain- 
ing unchanged,  but  to  obtain  a  given  current-carrying  capacity, 
or  a  small  enough  total  resistance,  the  total  cross  section  must 
not  be  lowered  below  a  certain  limit,  so  that,  in  general,  reducing 


308  Circular  of  the  Bureau  of  Standards 

the  diameter  of  the  strands  means  an  increase  in  the  number  of 
strands. 

With  enameled  strands  of  about  0.07  mm  bare  diameter  twisted 
together  to  form  a  composite  conductor  the  order  of  magnitude 
of  the  resistance  ratio  may  be  estimated  by  the  following  procedure. 
Calculate  by  Table  19,  page  311,  and  Table  17,  page  309,  the  resist- 
ance ratio  for  a  single  strand  at  the  desired  frequency  (this  value 
of  R/R0  will  lie  very  close  to  unity) ,  and  carry  out  the  same  calcu- 
lation for  the  equivalent  solid  wire,  whose  diameter  will  of  course 
be  d^fn,  where  n  =  the  number  of  strands  and  d  =  the  diameter 
of  a  single  strand.  Then  the  resistance  ratio  for  the  stranded 
conductor  will,  for  moderate  frequencies,  lie  about  one-quarter 
to  one-third  of  the  way  between  these  two  values,  being  closer  to 
the  lower  limit.  This  holds  for  straight  wires  up  to  higher  fre- 
quencies than  for  solenoids.  (See  critical  frequency  mentioned  in 
second  paragraph  below.)  Not  all  so-called  litzendraht  is  as  good 
as  this  by  any  means.  For  a  woven  tube  the  resistance  ratio  may 
be  as  low  as  one- tenth  of  the  way  from  the  lower  to  the  upper 
limits  mentioned. 

Coils  of  Stranded  Wire. — In  the  case  of  solenoids  wound  writh 
stranded  conductor,  the  resistance  ratio  is  always  larger  than  for 
the  straight  conductor,  and  at  high  frequencies  may  be  two  to 
three  times  as  great.  It  is  appreciably  greater  for  a  very  short 
coil  than  for  a  long  solenoid. 

For  moderate  frequencies  the  resistance  ratio  is  less  than  for  a 
similar  coil  of  solid  wire  of  the  same  cross  section  as  just  stated, 
but  for  every  stranded-conductor  coil  there  is  a  critical  frequency 
above  which  the  stranded  conductor  has  the  larger  resistance 
ratio.  This  critical  frequency  lies  higher  the  finer  the  strands 
and  the  smaller  their  number.  For  100  strands  of  say  0.07  mm 
diameter  this  limit  lies  above  the  more  usual  radio  frequencies. 

This  supposes  that  losses  in  the  dielectric  are  not  important, 
which  is  the  case  for  single-layer  coils  with  strands  well  insulated. 
In  multiple-layer  coils  of  stranded  wire;  dielectric  losses  are  not 
negligible  at  high  frequencies. 


Radio  Instruments  and  Measurements  309 

77.  TABLES  FOR  RESISTANCE  CALCULATIONS 
TABLE  17. — Ratio  of  High-Frequency  Resistance  to  the  Direct-Current  Resistance 

[See  formulas  (207),  (208),  and  (109)] 


z 

R 

R; 

Difference 

z 

R 
Ro 

Difference 

z 

R 
Ro 

Difference 

0 

1.0000 

0.0003 

5.2 

2.114 

0.070 

14.0 

5.209 

0.177 

0.5 

1.0003 

.0004 

5,4 

2.184 

.070 

14.5 

5.386 

.176 

.6 

1.0007 

.0005 

5.6 

2.254 

.070 

15.0 

5.562 

.353 

.7 

1.0012 

.0009 

5.8 

2.324 

.070 

.8 

1.  0021 

.0013 

6.0 

2.394 

.069 

16.0 

5.915 

0.353 

.9 

1.0034 

.0018 

6.2 

2.463 

.070 

17.0 

6.268 

.353 

18.0 

6.621 

.353 

1.0 

1.005 

0.003 

6.4 

2.533 

0.070 

19.0 

6.974 

.354 

1.1 

1.008 

.003 

6.6 

2.603 

.070 

20.0 

7.328 

.353 

1.2 

1.011 

.004 

6.8 

2.673 

.070 

1.3 

1.015 

.005 

7.0 

2.743 

.070 

21.0 

7.681 

0.353 

1.4 

1.020 

.006 

7.2 

2.813 

.071 

22.0 

8.034 

.353 

1.5 

1.026 

.007 

7.4 

2.884 

.070 

23.0 

8  387 

.354 

24.0 

8.741 

.353 

1.6 

1.033 

0.003 

7.6 

2.954 

0.070 

25.0 

9.094 

.353 

1.7 

1.042 

.010 

7.8 

3.024 

.070 

1.8 

1.052 

.012 

8.0 

3.094 

.071 

26.0 

9.447 

0.70 

1.9 

1.064 

.014 

8.2 

3.165 

.070 

28.0 

10.15 

.71 

2.0 

1.078 

.033 

8.4 

3.235 

.071 

30.0 

10.86 

.71 

32.0 

11.57 

.70 

2.2 

1.111 

0.041 

8.6 

3.306 

0.071 

34.0 

12.27 

.71 

2.4 
2.6 
2.8 
3.0 

1.152 
1.201 
1.256 
1.318 

.049 
.056 
.062 
.067 

8.8 
9.0 
9.2 
9.4 

3.376 
3.446 
3.517 
3.587 

.070 
.071 
.070 
.071 

36.0 
38.0 
40.0 
42.0 

12.98 
13.69 
14.40 
15.10 

0.71 
.71 
.70 
.71 

3.2 

1.385 

0.071 

9.6 

3.658 

0.070 

44.0 

15.81 

.71 

3.4 

1.456 

.073 

9.8 

3.728 

.071 

46.0 

16.52 

0.70 

3.6 

1.529 

.074 

10.0 

3.799 

.176 

48.0 

17.22 

.71 

3.8 
4.0 

1.603 
1.678 

.075 
.074 

10.5 
11.0 

3.975 
4.151 

.176 
.176 

50.0 
60.0 

17.93 
21.47 

3.54 
3.53 

4.2 

1.752 

0.074 

11.5 

4.327 

0.177 

70.0 

25.00 

3.54 

4.4 

1.826 

.073 

12.0 

4.504 

.176 

80.0 

28.54 

3.53 

4.6 

1.899 

.072 

12.5 

4.680 

.176 

90.0 

32.07 

3.54 

4.8 

1.971 

.072 

13.0 

4.856 

.177 

100.0 

35.61 

5.0 

2.043 

.071 

13.5 

5.033 

.176 

oo 

00 

310 


Circular  of  the  Bureau  of  Standards 


o 
o 

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V 


I 


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Frequency  •*-  10«  

Wave  length,  meters  

£ 

i 

^{lli  B 

s  J     1  1  11  1  f  1 

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Radio  Instruments  and  Measurements 


TABLE  19. — Values  of  the  Argument  «„  for  Copper  Wire  0.1  cm  Diameter  and 
Resistivity  1.724  Microhm-cms 


f 

cycles  per 
second 

at 

Difference 

X 
meters 

f 
cycles  per 
second 

a. 

Difference 

X 
meters 

100 

0  1071 

0.0443 

50  000 

2.395 

0.229 

6000 

200 

1514 

.0341 

60  000 

2  624 

.210 

5000 

300 

1855 

.0287 

70  000 

2.834 

.195 

4286 

400 

2142 

0253 

80  000 

3  029 

.184 

3750 

MB 

2395 

.0229 

90  000 

3.213 

.174 

3333 

600 

0.  2624 

0.0210 

100  000 

3.387 

0.761 

3000 

2834 

.0195 

150  000 

4  148 

.642 

2000 

800 

.3029 

.0184 

200  000 

4.790 

.565 

1500 

900 

.3213 

.0174 

250  000 

5.355 

.511 

1200 

3387 

.1403 

300  000 

5  866 

.318 

1000 

2000 

0.4790 

0.  1076 

333  333 

6.184 

0.380 

900 

3000 

5866 

.0908 

375  000 

6.564 

.452 

800 

4000 

6774 

.0799 

428  570 

7.012 

.561 

700 

•MAO 

7573 

.0723 

500  000 

7.573 

.723 

600 

6000 

0  82% 

0  0664 

600  000 

8.296 

.664 

500 

7000 

.8960 

.0619 

700  000 

8.960 

0.315 

429 

8000 

9579 

0581 

750  000 

9.275 

.304 

400 

9000 

1.0160 

.055 

800  000 

9.579 

.581 

375 

10  000 
15000 
20  000 
30  000 

1.071 
1.312 
1.514 
1  855 

0.241 
.202 
.341 
287 

30  000 
20  000 
15  000 
10  000 

900  000 
1  000  000 

1  500  000 
3  000  000 

10.16 
10.71 

13.12 
18.55 

.55 
2.41 

5.43 

333 
300 

200 
100 

40  000 

2.142 

.253 

7500 

TABLE  20.— Values  of  £  for  Use  with  Formula  (210) 


ft 

R 
R« 

Difference 

ft 

R 

c 

Difference 

ft 

R 
R. 

Difference 

o 

1.000 

1.0 

.086 

0.037 

2.5 

2.477 

0.111 

0.1 

1.000 

1.1 

.123 

.047 

2.6 

2.588 

.109 

.2 

1.000 

1.2 

.170 

.059 

2.7 

2.697 

.106 

.3 

1.001 

1.3 

'     .229 

.069 

2.8 

2.803 

.104 

.4 

1.002 

1.4 

.298 

.080 

2.9 

2.907 

.103 

.5 

1.006 

0.002 

1.5 

.378 

.090 

3.0 

3.010 

.101 

0.55 

1.008 

.004 

1.6 

1.468 

0.098 

3.1 

3.111 

0.101 

.60 

1.012 

.004 

1.7 

1.566 

.166 

3.2 

3.212 

099 

.65 

1.016 

.005 

1.8 

1.672 

.111 

3.3 

3.311 

.099 

.70 

1.021 

.007 

1.9 

1.783 

.115 

3.4 

3.410 

.099 

.75 

1.028 

.008 

2.0 

1.898 

.117 

3.5 

3.509 

.099 

0.80 

1.036 

0.009 

2.1 

2.015 

0.117 

3.6 

3.608 

0.098 

.85 

1.045 

.011 

2.2 

2.132 

.117 

3.7 

3.706 

.098 

.90 

1.057 

.013 

2.3 

2.248 

.115 

3.8 

3.804 

.098 

.95 

1.070 

.016 

2.4 

2.364 

.113 

3.9 

3.902 

.098 

1.00 

1.086 

2.5 

2.477 

.111 

4.0 

4.000 

312  Circular  of  the  Bureau  of  Standards 

MISCELLANEOUS  FORMULAS  AND  DATA 
78.  WAVE  LENGTH  AND  FREQUENCY  OF  RESONANCE 

Xcm  =  i  .8838  X  iou-Y/LC  (cgs  electromagnetic  units)  (214) 

=  6.283        VL  cgs  electromagnetic     C  cgs  electrostatic    (2  1  5) 


Am  =  0.0595  7    -\/L  cgs  electromagnetic     C  micromicrofarad  (216) 
=  1-884        VL  microhenry     C  micromicrofarad  (217) 


=  1884        VL  microhenry    C  microfarad  (2i8) 

=  5957°      VL  millihenry     C  microfarad  (219) 

=  i  884000  VL  henry     C  microfarad  (220) 


L  henry     C  microfarad 

5033 
millihenry     C  microfarad 


159  200 
VL  microhenry     C  microfarad 


i  OOP 


_  _ 
V-L  henry     C  microfarad 

_      31620 


millihenry     C  microfarad 


=  _  1  OOP  OOP  ___ 
VL  microhenry    C  microfarad 

T—  I  _27r 

~J~~^  (227) 

,     _  2.998  X  IP8 

Am—          —J  (228) 

_  1.  884X10*  ' 

~^~  (229) 


Radio  Instruments  and  Measurements  313 

79.  MISCELLANEOUS  RADIO  FORMULAS 

When  units  are  not  specified,  international  electric  units  are 
to  be  understood.  These  are  the  ordinary  units,  based  on  the 
international  ohm  and  ampere,  the  centimeter  and  the  second. 
Full  information  is  given  on  electric  units  in  reference  No.  152, 
Appendix  2. 

Current  in  Simple  Series  Circuit. — 
jl  E 

Phase  Angle. — 

=^  =  XLoXo  (23i) 


wL  --  p 

—  7  —  -  in  simple  series  circuit.  (232) 


Sharpness  of  Resonance.  — 


C 
Current  at  Parallel  Resonance. — 

ER 


/r2-/t» 

A3          _J ^  (2^\ 

E»,  .T"1  t>  \*3O/ 

£V-         KcoCr      K  (See  p.  3 7.) 


r>2    .   e,rt  (234) 

(See  p.  39.) 

Coefficient  of  Coupling.— 

(235) 

M 
.-   for  direct  and  inductive  coupling  (236) 


=     ^     '   for  capacitive  coupling.  (237) 

Cm  (See  p.  49.) 

Power  Input  in  Condenser  — 

P  =  0.5  X  io-WCE0*  watts  (238) 

for  C  in  microfarads,  E,  in  volts,  and  N  =  number  of  charges  per 
second. 


3*4  Circular  of  the  Bureau  of  Standards 

Power  Loss  in  Condenser  — 


Condenser  Phase  Difference  — 

(240) 
for  $  in  radians,  r  in  ohms,  C  in  farads. 

rC 
1^=0.1079  -^-degrees  (24I) 

for  r  in  ohms,  C  in  micromicrofarads,  X  in  meters. 

rC 
t  =  389-  y  seconds  (242) 

for  r  in  ohms,  C  in  micromicrofarads,  X  in  meters. 

.     o.ooi        X 
"*>  —  X  I^o  XO'I54  ohms  (243) 


for  ^  in  minutes,  C  in  microfarads,  X  in  meters. 
Energy  Associated  with  Inductance  — 

W  =  \LI'  (=44) 

Inductance  of  Coil  Having  Capacity: 

j  L 

*~i-u*CL  (245) 

for  C  in  farads,  L  in  the  denominator  in  henries. 

L.  =L  (i  +3.553  -jj-j-J  approximately  (246) 

for  X  in  meters,  C  in  micromicrofarads,  L  in  the  parentheses  in 
microhenries.    This  formula  is  accurate  when  the  last  term  is  small 
compared  with  unity. 
Current  Transformer  — 

A     n/I+o£A  (247) 

J,    n\      uLj  (Seep.  154.) 

Audibility  — 

(248) 
(See  p.  1  66.) 


Radio  Instruments  and  Measurements  315 

Natural  Oscillations  of  Horizontal  Antenna. — 

=  i,  3,  5,   (249) 


m 


for  X  in  meters,  C0  =  capacity  in  microfarads  for  uniform  voltage, 
L0  =  inductance  in  microhenries  for  uniform  current. 

Approximate  Wave  Length  of  Resonance  for  Loaded  Antenna. — 


37  (250) 

where  L  =  inductance  of  loading  coil  in  microhenries  and  other 
quantities  are  as  in  preceding  formula. 

Radiation  Resistance  of  an  Antenna. — 

(h  V 
T-  )  ohms  (2 si) 

*/ 

where  h  —  height  from  ground  to  center  of  capacity,  and  h  and  X 
are  in  the  same  units,  and  X  is  considerably  greater  than  the  fun- 
damental wave  length. 

Electron  Flow  From  Hot  Filament. — 

78  =  AT*6-|  (252) 

where  7B=  electron  current  in  milliamperes  per  centimeter 2  of  fila- 
ment surface,  T  =  absolute  temperature,  and  A  and  b  depend  on 
metal  of  filament;  for  tungsten  A  =  2.5  x  io10,  b  =  52500. 
Electron  Current  in  ^-Electrode  Tube. — 

T  ——\>    (  F*  ~\~  k  1)   )^  (/2.Xt'l\ 

B  \         B  1      I/  \      OO/ 

where  EB  =  plate  voltage,  z»1  =  grid  voltage,  &t  =  amplification  con- 
stant. 

Resistance  Measurement  by  Resistance — Variation  Method  Using 
Undamped  Emf. — 

R  =  R^j^rri  (254) 

Resistance  Measurement  by  Resistance — Variation  Method  Using 
Impulse  Excitation. — 

r>       r>  i  /^»^i-\ 

K  —  Kfj — ^-  (255) 

•      L  i 

Resistance  Measurement  by  Reactance-Variation  Method   Using 
Undamped  emf. — 

R=X'\IT^T'  (256) 

V  ^r        -*i 

where  X1  =  change  of  reactance  between  the  two  observations  of 
current.  Various  particular  cases  of  this  formula  are  given  in 
section  50. 


316  Circular  of  the  Bureau  of  Standards 

Natural  Frequency  of  Simple  Series  Circuit. — 

(257) 


CO  = : 

(258) 


Number  of  Oscillations  1o  Reduce  Current  to  i  Per  Cent  of  Initial 
Value  in  Wave  Train. — 

4.6 
n  =  -y  (259) 

Logarithmic  Decrement. — 

5  =  logey •  =  —  (260) 

A     7 

R 

=  7T — =-  = 


sharpness  of  resonance 

=  TT  X  phase  difference  of  condenser  or  coil,  the 
resistance  being  in  one  or  the  other 

average  energy  dissipated  per  cycle  _ 
2  X  average  magnetic  energy  at  the  current  maxima 

5=0.00167  -j-  (261) 

JLrf 

for  R  in  ohms,  X  in  meters,  L  in  microhenries. 

5  =  5918  -y-  (262) 

for  R  in  ohms,  X  in  meters,  C  in  microfarads. 

(263) 


for  R  in  ohms,  C  in  microfarads,  L  in  microhenries. 

Current  at  resonance  Produced  by  Slightly  Damped  emf  Induced  in 
a  Circuit.  — 

/v  F  - 

72  "*          "^  O  /       s-       \ 

=  i6/3L25'5(6'  +  5)  (264) 

Decrement  Measurement  by  Reactance  —  Variation  Method.  — 


(See  p.  187  for  variations  of  this  formula.) 


Radio  Instruments  and  Measurements 


317 


80.  PROPERTIES  OF  METALS 
TABLE  21 


Metal 

• 

Microhm- 
centimeters 
at  20°  C 

Temperature 
coefficient 
at  20°  C 

Specific 
gravity 

Tensile 
strength, 

Ibs./in.v 

Melting 
point, 
"C 

Advance.    See  Constantan. 
Aluminum  .....   .                   

2.828 
41.7 
120 

7 
7.6 

87 
49 
1.  7241 
1.771 

92 
33 

2.44 

10 

22 
4.6 
44 
95.783 
5.7 
42 
100 
7.8 
11 
7.8 
10 
1.59 
10.4 
11.9 
18 
70 

15.5 
47 
11.5 
5.6 
5.8 

0.0039 
.0036 
.004 
.002 
.0038 

.0007 
.00001 
.00393 
.00382 

.00016 
.0004 

.00342 

.0050 

.0039 
.004 
.00001 
.00089 
.604 
.0020 
.0004 
.006 
.0033 
.0018 
.003 
.0038 
.005 
.004 
.003 
.001 

.0031 

.00001 
.0042 
.0045 
.0037 

2.70 
6.6 
9.8 
8.6 
8.6 

8.1 
8.9 
8.89 
8.89 

8.9 
8.4 

19.3 

7.8 

11.4 
1.74 
8.4 
13.  546 
9.0 
8,9 
8.2 
8.9 
12.2 
8.9 
21.4 
10.5 
7.7 
7.7 
7.7 
7.5 

16.6 
8.2 
7.3 
19 
7.1 

30  000 

659 
630 
271 
900 
321 

1250 
1190 
1083 

Bismuth 

Brass    

70  000 

Cadmium  

Calido.    See  Nicbrome. 
Climax  

150  000 
120  000 
30  000 
60  000 

95  000 
150  000 

20000 

Constantan  

Copper,  annealed  

Eureka.    See  Constantan. 
Excello     

1500 
1100 

1063 

1530 

327 
651 
910 

-sag 

2500 
1300 
1500 
1452 
1550 
750 
1755 
960 
1510 
1510 
1510 
1260 

2850 

O^rman  silver,  1R  pe*  ^eft                  ......... 

German  silver,  30per  cent.    See  Constantan. 
Gold  

la  la.    See  Constantan. 
Ideal.    See  Constantan. 
Iron,  99.98  per  cent  pure        ... 

Iron.    See  Steel. 
Lead  

3  000 
33  000 
150  000 
0 

Magnesium  

Manganin  -  

Mercury    

Molybdenum,  drawn  

Monel  rnetnl  

160  000 
150  000 
120  000 
39  000 
25  000 
50  000 
42  000 
53  000 
58  000 
100  000 
230  000 

Nichrome  

Nickel    

Palladium  ,  ,  

Phosphor  bronze  

Platinum    

Silver  

Ftffl,  F   P  P                                    ^fe 

Steel,  B.B  

w* 
Steel,  Siemens-Martin  *.T*;i"' 

Pte^l,  T"angntx>Kp  .                       

Superior.    See  Climax. 
Tantalum  

Therlo  

Tin  

4000 
500  000 
10  000 

232 
3000 
419 

Tungsten,  drawn  

Zinc  

The  resistivities  given  in  Table  21  are  values  of  p  in  the  equa- 
tion R0  =  p-,  where  /  =  length  in  centimeters  and  s  =  cross  section  in 
s 

square  centimeters .  This  formula  gives  the  low-frequency  or  direct- 
current  resistance  of  a  conductor.  For  the  calculation  of  resist- 
ances at  high  frequencies,  see  Tables  17  to  20,  pages  309-311. 


318  Circular  of  the  Bureau  of  Standards 

The  values  given  for  resistivity  and  temperature  coefficient  of 
copper  are  the  international  standard  values  for  commercial 
copper.  Any  departure  from  this  resistivity  is  accompanied  by 
an  inverse  variation  in  the  temperature  coefficient.  This  is  true 
in  a  general  way  for  other  metal  elements.  In  tjie  case  of  copper 
the  resistivity  and  temperature  coefficient  are  inversely  propor- 
tional, to  a  high  degree  of  accuracy. 

The  "temperature  coefficient  at  2o°C"  is  a20  in  the  equation 
Rt  =  R2Q  (i  +a20[^  — 20]).  In  some  cases  the  temperature  variation 
does  not  follow  a  straight-line  law;  in  such  cases  a20  applies  only 
to  a  small  range  of  temperature  close  to  20°.  Steel  is  an  example, 
the  resistance  rise  at  high  temperatures  being  faster  than  propor- 
tional to  temperature. 

Constantan  and  the  other  wires  (Advance,  etc.)  having  substan- 
tially the  same  properties,  are  alloys  of  approximately  60  per  cent 
copper  and  40  per  cent  nickel.  They  are  used  in  rheostats  and 
measuring  instruments. 

German  silver  is  an  alloy  of  copper,  nickel,  and  zinc.  The  per 
cent  stated  indicates  the  percentage  of  nickel. 

Manganin  contains  about  84  per  cent  copper,  12  per  cent  man- 
ganese, and  4  per  cent  nickel.  It  is  the  usual  material  in  resist- 
ance coils.  Its  very  small  thermal  electromotive  force  against 
copper  is  one  of  its  main  advantages.  The  similar  alloy,  therlo, 
is  used  for  the  same  purposes. 

Monel  metal  is  an  alloy  containing  approximately  71  per  cent 
nickel,  27  per  cent  copper,  and  2  per  cent  iron. 


Bureau  of  Standards  Circular  No.  74 


FIG.  214. — Variable  condensers  used  as  standards  of  capacity 


FIG.  215. — Single-layer  coils  used  as  standards  of  inductance 


Bureau  of  Standards  Circular  No.  74 


FIG.  216. — Multiple-layer  standard  coil 


FIG.  217. — Standard  wave  length  circuit 


APPENDIXES 


APPENDIX  1.—  RADIO  WORK  OF  THE  BUREAU  OF  STANDARDS 

The  functions  of  the  radio  laboratory  of  this  Bureau  include  the  maintenance  of 
standards  for  radio  measurements,  the  testing  of  instruments  and  apparatus,  technical 
assistance  in  radio  matters  to  various  branches  of  the  Government,  and  researches  in  the 
theory  and  practice  of  radio  communication.  The  activities  of  the  Bureau  in  some  of 
these  lines  have  been  to  a  considerable  extent  covered  in  the  foregoing  sections. 
A  more  comprehensive  account  is  given  here  of  the  facilities,  accomplishments,  and 
aims  of  this  laboratory. 

This  account  does  not  include  a  description  of  the  work  of  the  United  States  naval 
radiotelegraphic  laboratory  or  of  the  Signal  Corps  laboratory,  both  of  which  are  located 
at  the  Bureau  of  Standards.  They  were  in  existence  before  the  Bureau's  own  radio 
laboratory  was  established,  and  the  publications  of  the  Naval  Laboratory  are  printed 
in  the  Bulletin  of  the  Bureau  of  Standards.  A  list  of  these  publications  is  given  in 
the  "Bibliography,"  page  329. 

1.  DEVELOPMENT  AND  MAINTENANCE  OF  STANDARDS 

Capacity.  —  The  qualities  desirable  in  a  condenser  to  be  used  as  a  standard  at  radio 
frequencies  are:  Constancy  of  capacity  with  varying  frequency  and  temperature 
and  other  conditions,  small  resistance  or  phase  difference,  careful  shielding,  and  con- 
venience of  design.  The  quartz-pillar  air  condensers  described  above  (p.  120)  have 
these  qualities  and  are  satisfactory  fundamental  standards  for  radio  measurements. 
They  are  the  result  of  many  years'  experience  at  this  Bureau  in  the  measurement  and 
design  of  condensers.  The  laboratory  has  a  set  of  variable  condensers  of  this  type, 
with  maximum  capacities  ranging  from  o.oooi  to  0.0075  microfarad.  Having  con- 
tinuously variable  capacity,  they  are  very  convenient  to  use  in  a  standard  circuit. 
Fixed-value  condensers  of  the  same  general  type  with  greater  capacity  are  also  used. 
Good  mica  condensers,  well  made  and  properly  shielded,  may  also  be  used  as  standards 
at  radio  frequencies.  The  best  mica  condensers  have  lower  phase  differences  than 
many  air  condensers  of  ordinary  design,  because  of  the  solid  dielectric  used  to  insulate 
the  plates  in  the  air  condensers.  Fixed-value  condensers  for  radio  use  should  pref- 
erably be  independent  and  not  parts  of  a  permanently  connected  set  of  condensers, 
on  account  of  mutual  capacities  between  the  parts  of  such  a  set. 

The  capacities  of  air  condensers  used  as  radio  standards  are  determined  by  low- 
frequency  measurements,  either  by  the  absolute  Maxwell  bridge  method  38  at  a  fre- 
quency of  too  per  second,  or  by  alternating-current  comparison  37  with  standard  con- 
densers at  frequencies  from  100  to  3000.  The  plate  and  lead  resistances  and  inductances 
of  these  condensers  are  negligibly  small,  the  insulation  resistance  is  extremely  high, 
and  the  phase  difference  due  to  absorption  is  very  small.  A  few  of  the  condensers 
have  a  phase  difference  which  can  just  be  detected  at  low  settings.  It  is  so  small  as  to 
cause  no  change  of  capacity,  as  shown  by  agreement  of  the  capacities  at  low  and 
high  settings  at  different  frequencies.  These  condensers  have  practically  zero 
temperature  coefficient  and  have  remained  constant  in  capacity. 

84  See  reference  No.  174,  Appendix  i.  r  See  reference  No.  176,  Appendix  a. 

3*9 
35601°—  18  -  21 


320  Circular  of  the  Bureau  of  Standards 

Inductance. — The  problem  of  developing  standards  of  inductance  for  use  at  radio 
frequencies  is  mainly  that  of  minimizing  resistance  and  distributed  capacity.  These 
requirements  are  both  met  by  the  single-layer  coil  for  inductances  of  moderate 
value.  The  shape  of  coil  having  the  minimum  length  of  wire  (and  hence  minimum 
resistance  if  the  cross  section  of  the  wire  is  specified)  for  a  required  inductance  is 
given  on  pages  286  to  292.  The  capacity  of  a  coil  is  roughly  proportional  to  the  radius 
of  the  coil  and  independent  of  the  number  of  turns  and  length.  For  a  single-layer 
coil  having  a  close  winding  the  value  in  micromicrofarads  is  approximately  equal  to 
the  numerical  value  of  the  diameter  in  centimeters.  For  a  single-layer  coil  with 
spaced  winding  on  an  open  form,  like  those  described  below,  the  capacity  in  micro- 
microfarads  is  approximately  equal  to  the  numerical  value  of  the  radius  in  centi- 
meters. Hence  a  coil  should  be  made  longer  and  of  smaller  diameter  than  the  theo- 
retical shape  indicated  for  minimum  resistance  in  order  to  reduce  the  capacity. 

The  single-layer  coils  used  as  standards  in  this  laboratory  conform  to  these  prin- 
ciples. Theset  shown  in  Fig.  215,  facing  page3i8,  ranges  in  diameter  from  13  1x>38cm 
and  in  inductance  from  60  to  5350  microhenries.  The  capacities  of  the  coils  with 
their  leads  range  from  9  micromicrofarads  for  the  smallest  to  16  for  the  largest.  The 
capacities  are  kept  small  by  eliminating  as  much  dielectric  as  possible  from  the  neigh- 
borhood of  the  wire.  The  coils  are  wound  with  silk-covered  "litzendraht",  with  the 
turns  spaced.  The  open  form  gives  the  coils  the  shape  of  a  i2-sided  polygon  instead 
of  a  circle. 

Multiple-layer  coils  are  used  as  standards  for  inductances  larger  than  any  of  these 
single-layer  standards.  These  are  satisfactory  if  the  wires  are  spaced  well  apart  and 
the  amount  of  dielectric  between  the  turns  and  layers  is  kept  small.  On  this  account 
it  is  not  desirable  to  impregnate  such  coils  with  insulating  compound.  Such  a  coil 
is  shown  in  Fig.  216,  facing  page  319. 

The  inductances  of  the  standard  coils  are  determined  by  intercomparison  in  circuits 
using  the  standard  air  condensers  referred  to  above.  The  basis  of  these  intercompari- 
sons  is  a  rectangular  inductance  consisting  of  a  single  turn  of  copper  tubing.  The 
inductance  of  this,  obtained  by  calculation,  is  9  microhenries.  For  discussion  of 
these  determinations  see  page  247. 

Wave  Length. — The  wave-length  standards  consist  essentially  of  standard  circuits 
made  up  of  the  standard  condensers  and  inductance  coils  just  described.  As  shown 
in  Fig.  217,  facing  page  313,  the  circuit  includes  a  pair  of  leads;  the  inductance  coil  is 
considered  to  include  these  when  its  value  is  determined.  A  wire  leading  to  ground  is 
connected  to  the  shielded  side  of  the  condenser.  A  current  indication  is  obtained  in  a 
thermal  ammeter  in  a  separate  circuit  near  the  standard  circuit.  This  separate  circuit 
is  placed  by  trial  at  such  a  distance  that  it  does  not  affect  appreciably  the  capacity  or 
inductance  of  the  standard  circuit.  An  alternative  method  of  observing  the  current 
in  the  standard  circuit  is  the  use  of  a  thermoelement  in  series,  the  circuit  being  stand- 
ardized with  the  thermoelement  in. 

The  range  of  wave  lengths  obtained  with  the  coils  and  condensers  described  above 
is  from  100  to  13  ooo  meters. 

The  work  of  the  Bureau  in  connection  with  wave-length  standardization  includes 
also  the  development  of  the  decremeter  described  on  pages  196-199.  This  instru- 
ment is  a  wave  meter  which  has  a  more  nearly  uniform  scale  of  wave  lengths  than  wave 
meters  employing  the  ordinary  condenser  with  semicircular  plates.  It  is  built  for 
wave  lengths  from  75  meters  up. 

Current. — High-frequency  current  standardization  is  at  present  based  upon  thermal 
ammeters.  For  small  currents  the  standard  instruments  are  thermoelements.  These 
are  made  of  such  fine  wire  that  the  resistance  does  not  change  with  frequency.  When 
made  with  a  resistance  of  i  ohm  or  less,  a  thermoelement  may  be  inserted  directly  in 
a  radio  circuit  without  reducing  the  current  materially. 


Bureau  of  Standards  Circular  No.  74 


FIG.  219. — Small-typj  decremeter 


IG.  220. — Navy-type  decremeter 


Radio  Instruments  and  Measurements  321 

For  currents  of  intermediate  value,  a  hot-wire  ammeter  with  a  single  wire  is  taken 
as  a  standard.  The  principal  precaution  necessary  is  that  the  heated  wire  be  fine 
enough  to  remain  constant  in  resistance  at  all  frequencies  used. 

For  measurements  of  large  currents,  instruments  with  multiple  wires  or  strips  are 
taken  as  standards,  in  which  careful  investigation  has  shown  that  the  current  distri- 
bution among  the  wires  or  strips  does  not  change  with  frequency.  Ammeters  of  the 
cylindrical  type  with  a  thermocouple  on  each  wire  or  strip  have  been  developed  for 
this  purpose.  The  questions  of  errors  and  design  are  treated  above,  section  41.  The 
standard  instruments  thus  far  developed  are  suitable  for  measuring  currents  up  to  50 
amperes  at  frequencies  up  to  i  ooo  ooo. 

Resistance  and  decrement. — The  standards  used  in  the  determination  of  resistance 
and  decrement  are  of  two  classes.  In  the  first  class,  in  which  resistance  is  measured 
by  the  substitution  or  the  deflection  method,  resistance  standards  of  manganin  wire 
are  used.  These  are  short,  straight  lengths  of  fine  wire,  the  substitution  of  which  in  a 
circuit  does  not  appreciably  change  its  inductance.  (See  p.  178-180  for  further 
details.) 

In  the  second  class  of  measurement,  the  determination  of  resistance  depends  ulti- 
mately on  the  reactance  of  a  circuit  and  the  deflections  of  an  ammeter.  From  the 
variation  of  reactance  required  to  produce  a  certain  change  of  current  in  the  circuit 
the  resistance  is  obtained.  In  the  decremeter  described  on  page  196,  which  was  devel- 
oped at  this  Bureau,  the  decrement  is  obtained  directly  by  manipulation  of  the 
instrument  without  the  necessity  of  any  calculation.  A  dial  is  graduated  in  terms 
of  decrements  from  o  to  0.3  readable  to  o.ooi.  Photographs  of  several  types  of  the 
instrument  are  given  in  Figs.  218  to  220. 

2.  TESTING  OF  INSTRUMENTS  AND  MATERIALS 

Most  of  the  radio  apparatus  which  the  Bureau  is  called  on  to  test  is  standardized  by 
direct  comparison  with  the  standards  described  in  the  preceding  section.  The  fees 
which  have  been  established  for  testing  radio  and  other  electrical  apparatus,  and 
instructions  to  applicants  for  tests,  are  given  in  this  Bureau's  Circular  No.  6,  Fees  for 
Electric,  Magnetic,  and  Photometric  Testing.  For  tests  not  listed  there,  a  special  fee 
of  nominal  amount  is  charged.  Unless  otherwise  specified,  apparatus  is  tested  with 
undamped  current  using  a  pliotron  as  the  source.  Such  current  is  very  steady  and 
gives  the  maximum  accuracy  of  measurement.  In  general,  the  Bureau  does  not 
certify  an  accuracy  better  than  i  per  cent  on  any  radio  apparatus. 

Wave  meters. — A  wave  meter  is  tested  by  direct  comparison  with  a  standard  circuit, 
both  being  coupled  to  the  same  source  of  high-frequency  current.  For  the  procedure 
see  section  30.  If  a  ground  connection  is  to  be  used  on  the  wave  meter,  it  is  tested 
with  a  ground  on.  It  is  usually  most  convenient  to  use  an  ordinary  commercial  wave 
meter  without  a  ground  connection.  Grounding  makes  very  little  difference  in  the 
indications  of  the  instrument,  except  in  wave  meters  where  the  condenser  has  unusu- 
ally small  capacity  at  the  low  settings.  Wave  meters  which  are  to  be  used  as  instru- 
ments of  precision  are  tested  at  the  points  specified  by  the  applicant  for  test. 

Coils. — An  inductance  coil  is  tested  by  a  substitution  method,  other  coils  of  nearly 
the  same  value  being  substituted  in  a  circuit  which  is  tuned  to  resonance  by  a  variable 
condenser.  By  varying  the  capacity  of  the  condenser,  or  if  necessary  inserting  addi- 
tional inductance,  the  wave  length  is  varied,  and  a  curve  of  inductance  against  wave 
length  may  be  plotted.  Such  a  curve  is  shown  on  page  64. 

Condensers. — Condensers  are  also  tested  by  substitution,  either  by  placing  the  test 
condenser  and  the  standard  successively  in  the  same  position  or  by  using  a  double- 
throw  switch.  The  use  of  variable  standard  condensers  makes  this  measurement 
very  simple. 

Ammeters. — High-frequency  ammeters  are  standardized  by  comparison  with  a 
Standard  ammeter  in  series  in  the  same  circuit.  Test  is  usually  made  at  more  than 


322  Circular  of  the  Bureau  of  Standards 

one  wave  length.  No  regular  fees  have  been  established  for  this  as  yet,  each  test  being 
subject  to  a  special  fee  depending  on  the  time  consumed  in  the  measurement. 

Resistance  Measurements. — The  resistances  of  high-frequency  resistance  standards, 
of  wires  or  other  conductors,  and  of  coils,  condensers,  or  circuits,  are  measured  in  any 
of  the  ways  mentioned  in  sections  47  to  50  above.  These  include  substitution  or 
deflection  methods  in  terms  of  standards,  and  the  reactance-variation  methods.  In 
the  case  of  stranded  wire  submitted  for  high-frequency  resistance  measurement,  it  is 
desirable  to  measiire  also  the  low-frequency  or  direct-current  resistance  of  each  sepa- 
rate strand  and  the  insulation  resistance  between  strands. 

Insulating  Materials. — Tests  are  made  of  the  dielectric  loss  or  phase  difference  of 
insulating  materials  if  submitted  in  large  thin  sheets.  Measurements  of  dielectric 
strength  are  made  with  low-frequency  voltage  up  to  100  ooo  volts.  An  equipment  has 
also  been  developed  for  voltage  tests  at  radio  frequencies  up  to  20  ooo  volts. 

Operating  Apparatus. — Complete  transmitting  and  receiving  sets,  accessories,  and 
parts  of  sets  are  tested  when  the  circumstances  render  the  test  of  such  importance  as 
to  justify  the  Bureau  in  undertaking  the  work.  Tests  of  the  performance  of  complete 
sets  have  not  yet  been  standardized,  as  each  set  submitted  presents  a  distinct  prob- 
lem. Such  a  test  may  include:  Output  of  transmitter,  wave  forms  of  current  and 
voltage  of  power  supply  circuits,  purity  and  decrement  of  generated  wave,  wave 
lengths  of  transmitter  and  receiver,  selectivity  and  sensitivity  of  receiver. 

3.  RADIO  ENGINEERING  FOR  THE  GOVERNMENT 

The  testing  and  research  work  is  of  direct  value  to  the  Army,  Navy,  and  various 
other  branches  of  the  Government,  but  in  addition  to  this  the  laboratory  performs 
special  services  for  Government  Bureaus,  in  particular  those  of  the  Department  of 
Commerce.  Some  of  the  special  lines  of  work  thus  pursued  are  described  below. 
Technical  information  is  also  furnished  upon  request.  The  subjects  upon  which 
information  has  thus  been  furnished  include:  The  installing  of  transmitting  and 
receiving  equipment,  the  efficiency  of  radio  apparatus,  the  adjustment  of  equipment 
to  comply  with  the  law,  the  design  of  measuring  instruments,  formulas,  and  data. 
Assistance  is  rendered  the  Government  in  the  preparation  of  legislation  on  radio 
matters. 

Design  of  Instruments. — Portable  testing  equipments  have  been  developed  for  the 
radio  inspectors  of  the  Bureau  of  Navigation  of  the  Department  of  Commerce.  The 
decremeter  and  the  voltammeter  for  this  purpose  (described  in  preceding  sections) 
were  designed,  construction  supervised,  and  calibrated  here.  Technical  problems 
in  connection  with  instruments,  which  have  arisen  in  the  radio  inspections,  have 
been  referred  to  this  laboratory  for  solution. 

Design  of  Radio  Sets. — Complete  radio  transmitting  and  receiving  sets  have  been 
designed  and  furnished  to  three  of  the  Bureaus  of  the  Department  of  Commerce. 
These  are  in  use  on  the  ships  of  the  Lighthouse  Service,  Bureau  of  Navigation,  and 
Coast  and  Geodetic  Survey.  The  transmitters  are  built  in  compact  panel  form  and 
are  supplied  with  i  kw  of  power  in  a  motor  generator  delivering  5oo-cycle  current. 
This  current  flows  in  a  closed-core  transformer,  adjusted  for  maximum  efficiency,  to 
the  secondary  of  which  are  connected  a  quenched  gap  of  special  design,  mica  con- 
densers, and  a  flat  spiral  coil.  A  simple  switch  sets  the  wave  length  on  600,  750,  and 
1000  meters.  The  sets  handle  relatively  little  traffic,  and  have  a  range  of  about  260 
km.  Two  views  of  the  transmitter  are  shown  in  Figs.  221  and  222,  facing  page  322. 
The  transformer  and  the  inductance  spirals,  which  were  given  special  attention  in 
the  development  work,  are  shown  in  Fig.  223,  facing  page  323. 

The  receiver  designed  for  these  sets  consists  of  two  circuits,  the  antenna  circuit 
and  a  closed  detecting  and  measuring  circuit  inductively  coupled  to  it.  The  closed 
circuit  and  the  antenna  loading  coils  and  variable  condenser  are  all  contained  in  a 


Bureau  of  Standards  Circular  No.  74 


FIG.  221. — Transmitting  set  designed  by  Bureau  FIG.  222. — Transmitting  set  designed  by  Bureau 

of  Standards  (front  -view)  of  Standards  (rear  view) 


Bureau  of  Standards  Circular  No.  74 


FIG.  175. — Quenched  gap  plate  showing  the  circular  silver  sparking  surface 


FIG.  223. — Inductance  spirals  and  transformer  used  in  the 
transmitting  set  shown  in  Figs.  221  and  222 


FIG.  224. — Receiving  set  designed  by  Bureau  of  Standards 


Radio  Instruments  and  Measurements  323 

compact  cabinet.  The  closed  circuit  includes  a  variable  condenser  of  the  decremeter 
type,  and  serves  as  a  wave  meter  and  decremeter  as  well  as  acting  as  a  receiver  by 
virtue  of  the  crystal  detector  connected  across  the  condenser.  The  receiver  may  be 
tuned  to  wave  lengths  from  about  500  to  2500  meters.  Two  views  are  shown  in  Fig. 
224,  facing  page  323. 

Fog  Signaling  Apparatus. — The  Bureau  of  Standards  has  been  active  in  its  efforts 
to  promote  safety  at  sea  by  means  of  radio  apparatus.  An  equipment  was  designed 
and  constructed  for  use  at  a  lighthouse,  which  should  efficiently  supplement  the  light 
of  a  lighthouse  during  fog  and  prove  of  great  assistance  to  navigation.  An  auto- 
matic transmitting  device  is  arranged  to  send  out  a  characteristic  signal  once  every 
minute  on  a  short  wave  length,  so  that  it  will  be  readily  received  by  all  ships  within 
a  few  miles  of  the  lighthouse.  A  direction  finder  was  developed  for  use  on  ships 
receiving  the  signal,  so  that  they  can  get  their  bearings  by  radio. 

Field  Work. — Inspection  and  other  trips  are  made  at  the  request  of  other  Govern- 
ment bureaus.  Assistance  has  thus  been  rendered  to  the  Bureau  of  Navigation  of 
the  Department  of  Commerce  in  order  to  solve  technical  problems  that  have  arisen 
in  the  radio  inspections.  Such  problems  have  included  the  equipment  of  emer- 
gency radio  sets  on  shipboard,  cases  of  interference,  use  of  instruments  and  testing 
equipment,  etc. 

4.  RESEARCH  WORK. 

Military  Researches. — The  testing  of  instruments  and  materials  is  of  direct  or  indi- 
rect benefit  to  the  military  departments.  Additional  service  of  military  value  is 
being  rendered  by  the  laboratory  through  the  results  of  most  of  the  investigations 
which  are  in  progress.  These  investigations  are  of  both  a  scientific  and  engineering 
nature.  It  is  obviously  impossible  to  publish  any  description  of  this  work. 

Radio  Instruments  and  Methods  of  Measurement. — A  number  of  problems  in  radio 
measurements  are  being  studied  in  the  laboratories  of  the  Bureau  of  Standards. 
Some  of  these  have  been  brought  to  the  point  where  a  publication  has  been  issued  or 
a  testing  routine  established,  but  all  of  them  remain  fruitful  fields  for  investigation. 
Among  the  more  important  problems  is  that  of  establishing  wave-length  standards. 
The  standard  circuits  which  have  been  developed  are  described  above.  The  pro- 
duction and  measurement  of  large  currents  and  high  voltages  is  another  branch  of  the 
work.  In  this  connection  one  publication  has  been  issued,  Scientific  Paper  No.  206, 
"High-Frequency  Ammeters,"  and  a  special  type  of  volt-ammeter  has  been  designed. 
These  investigations  have  shown  that  simplicity  of  circuit  is  a  great  desideratum  for 
many  radio  measurements. 

The  measurement  of  resistance  and  decrement  has  received  considerable  attention. 
A  number  of  methods  have  been  used,  and  their  limitations  studied.  An  apparatus 
for  quick  measurements  has  been  developed;  it  is  described  in  Scientific  Paper  No. 
235,  "A  Direct-Reading  Decremeter  for  Measuring  the  Logarithmic  Decrement  and 
Wave  Length  of  Electromagnetic  Waves." 

Properties  of  Conductors  and  Insulators. — Data  are  obtained  on  the  ratio  of  high- 
frequency  to  low-frequency  resistance  of  stranded  wire  of  various  kinds.  This  work 
may  be  extended  to  strips,  tubes,  and  other  special  forms  of  conductors.  Insulating 
materials  are  studied  for  dielectric  loss,  dielectric  constant  and  its  variation  with 
frequency,  surface  flashover  voltage,  etc.  There  is  great  need  for  systematic  study 
both  of  the  methods  of  measurement  and  of  the  properties  of  these  materials. 

Inductance  Coils. — The  capacity  and  the  resistance  of  radio  coils  and  their  effect 
upon  the  inductance  furnish  an  interesting  problem.  The  effects  of  varying  shape, 
size,  pitch  and  size  and  kind  of  conductor,  insulation  of  conductor,  and  material  and 
kind  of  mounting,  all  require  investigation,  as  well  as  the  modes  of  connection  to 
radio  coils  and  the  effects  produced  by  combinations  of  coils. 

Electron  Tubes. — The  characteristics  and  applications  of  three-electrode  thermionic 
tubes  constitute  a  most  important  field  of  investigation.  These  tubes  have  been 


324  Circular  of  the  Bureau  of  Standards 

found  to  be  excellent  sources  of  current  for  laboratory  measurements.  A  number  of 
applications  to  military  uses  are  under  development.  The  characteristic  curves  of 
tubes  are  studied,  and  different  types  of  tubes  compared  as  amplifiers,  generators, 
and  detectors.  Special  attention  is  given  to  the  production  of  maximum  current  in 
generating  circuits  for  particular  purposes,  and  to  the  modulation  of  the  radio- 
frequency  current. 

Antennas. — Some  of  the  great  variety  of  problems  presented  by  the  antenna  are 
under  study.  The  properties,  functioning,  and  merits  of  antennas  of  various  forms 
for  particular  purposes  are  investigated.  The  means  of  supplying  current  to  the  an- 
tenna are  studied.  The  investigation  includes  the  consideration  of  the  behavior  and 
transmission  of  the  electromagnetic  waves  emitted  from  an  antenna.  Measurements 
of  antenna  resistance,  inductance,  and  capacity  are  made.  One  publication  has  been 
issued,  Scientific  Paper  No.  269,  "  Effect  of  Imperfect  Dielectrics  in  the  Field  of  a 
Radiotelegraphic  Antenna." 

APPENDIX  2.— BIBLIOGRAPHY 

This  bibliography  is  by  no  means  comprehensive.  A  few  of  the  more  important 
references  are  given  for  each  of  the  subjects  treated  in  the  text.  In  many  of  the  publi- 
cations listed  here  references  are  given  to  previous  publications.  Bibliographies  of 
the  current  literature  have  been  given  bimonthly  in  the  "Jahrbuch  der  drahtlosen 
Telegraphic"  since  1907.  Articles  on  radio  measurements  as  well  as  other  phases  of 
radio  communication  appear  in  the  bimonthly  ' '  Proceedings  of  the  Institute  of  Radio 
Engineers." 

ELEMENTARY  ELECTRICITY. 

1.  Elementsof  Electricity  and  Magnetism,  J.  J.  Thomson;4th  ed..  1909  (Cambridge). 

2.  Modern  Views  of  Electricity,  O.  J.  Lodge;  1889  (MacMillan). 

3.  The  Elements  of  Physics,  Vol.  II,  Electricity  and  Magnetism,  Nichols  and 

Franklin;  1905  (MacMillan). 

4.  Electricity  and  Magnetism,  R.  T.  Glazebrook;  1910  (Cambridge). 

5.  Elements  of  Electricity  for  Technical  Students,  W.  H.  Timbie;  1911   (John 

Wiley  &  Sons). 

6.  Magnetism  and  Electricity  for  Students,  H.  E.  Hadley;  1910  (MacMillan). 

7.  The  Elements  of  Electricity  and  Magnetism,  Franklin  and  MacNut;  1914  (Mac- 

Millan). 

8.  Elementary  Lessons  in  Electricity  and  Magnetism,  S.  P.  Thompson;  7th  ed., 

1915  (MacMillan). 

9.  A  Treatise  on  Electricity,  F.  B.  Pidduck;  1916  (Cambridge). 

ga.  Electricity  and  Magnetism,  S.  G.  Starling;  1912  (Longmans,  Green  &  Co.). 

ATLERNATING  CURRENTS. 

•> 

11.  Alternating  Currents,  Bedell  and  Crehore;  4th  ed.,  1901  (McGraw-Hill). 

12.  Alternating  Currents  and  Alternating  Current  Machinery,  D.  C.  and  J.  P.  Jack- 

son; 1896  (MacMillan). 

13.  The  Theory  of  Alternating  Currents  (2  vols.),  A.  Russell:  2d  ed.,  1914  (Cam- 

bridge). 

14.  Kapazitat  tmd  Induktivitat,  E.  Orlich;  1909. 

15.  Calculation  of  Alternating  Current  Problems,  L.  Cohen;  1913  (McGraw-Hill). 

16.  The  Foundations  of  Alternating  Current  Theory,  C.  V.  Drysdale;  1910  (E.  Ar- 

nold). 

17.  Transient  Electric  Phenomena  and  Oscillations,  C.  P.  Steinmetz;  1909  (McGraw- 

Hill). 


Radio  Instruments  and  Measurements  325 

COUPLED  CIRCUITS. 

21.  Currents  in  Coupled  Circuits;  A.  Oberbeck;  Annalen  der  Physik,  291,  p.  623; 

1895. 

22.  Use  of  Coupled  Circuits;  F.  Braun;  Physikalische  Zs.,  3,  p.  148;  1901. 

23.  Coupling  phenomena;  M.  Wien;  Annalen  der  Physik,  61,  p.  151,  1897;  25,  p.  i, 

1908. 

24.  Maximum  Current  in  the  Secondary  of  a  Transformer;  J.  S.  Stone;  Physical 

Review,  32,  p.  399;  1911. 

25.  Cisoidal  Oscillations;  G.  A.  Campbell;  Trans.  A.  I.  E.  E.,  30,  p.  873;  1911. 

26.  The  Impedances,  Angular  Velocities,  and  Frequencies  of  Oscillating-Current 

Circuits;  A.  E.  Kennelly;  Proc.  I.  R.  E.  4,  p.  47;  1916. 

27.  Alternating  and  Transient  Currents  in  Coupled  Electrical  Circuits;  F.  E.  Pernot; 

University  of  California,  publications  in  Engineering,  1,  p.   161;  1916. 

28.  Oscillograph  Demonstrations  of  Coupled  Circuits;  G.  W.  O.  Howe;  Proc.  Physical 

Society  London,  23,  p.  237;  1911.     J.A.Fleming;  Proc.  Physical  Society  Lon- 
don, 25,  p.  217;  1913. 

29.  Mechanical  Models;  T.  R.  Lyle;  Phil.  Mag.,  25,  p.  567;  1913.    W.  Deutsch; 

Physikalische  Zs.,  16,  p.  138;  1915. 

ANTENNA  CALCULATIONS. 

31.  Theory  of  Horizontal  Antennas;  J.  S.  Stone;  Trans.  Int.  Elec.  Congress,  St. 

Louis,  3,  p.  555;  1904. 

32.  Theory  of  Loaded  Antenna;  A.  Guyau;  La  Lumiere  Electrique,  15,  p.  13;  1911. 

33.  Capacity  of  Radiotelegraphic  Antennas;  G.  W.  O.  Howe;  Electrician,  73,  pp. 

829,  859,  906,  1914;  75,  p.  870;  1915. 

34.  The  Electrical  Constants  of  Antennas;  L.  Cohen;  Elec.  World,  65,  p.  286;  1915. 

DAMPING. 

41.  Theory  of  Free  Oscillations;  Alternating  Current  Phenomena,  C.  P.  Steinmetz; 

Appendix  II,  p.  709;  4th  ed.,  1908. 

42.  Decrements  in  Coupled  Circuits;  V.  Bjerkhes;  Annalen  der  Physik,  44,  pp. 

74,  92,  1891;  291,  p.  121,  1895.     M.  Wien,  Annalen  der  Physik,  25,  p.  625, 
1908;  29,  p.  679,  1909. 

43.  Linear  Decrement:  J.  S.  Stone;  Electrician,  73,  p.  926;  1914.     Proc.  I.  R.  E-, 

2,  p.  307,  1914;  4,  p.  463,  1916. 

ELECTROMAGNETIC  WAVES. 

51.  A  Treatise  on  Electricity  and  Magnetism;  J.  C.  Maxwell;  1873. 

52.  Recent  Researches  in  Electricity  and  Magnetism;  J.  J.  Thomson;  1893. 

53.  Electromagnetic  Theory  (3  vols.);  O.  Heaviside;  1893. 

54.  Signaling  Through  Space  Without  Wires;  O.  J.  Lodge;  1894. 

55.  Derivation  of  Equations  of  a  Plane  Electromagnetic  Wave;  E.  B.  Rosa;  Phys. 

Rev.,  8,  p.  282;  1899. 

56.  Electric  Waves;  H.  Hertz  (translated  into  English  by  D.  E.  Jones);  1900. 

57.  Maxwell's  Theory  and  Wireless  Telegraphy;  H.  Poincare  (translated  into  Eng- 

lish by  F.  K.  Vreeland) ;  1904. 

58.  Researches  in  Radiotelegraphy;  J.  A.  Fleming;  Smithsonian  Report  for  1909, 

P-  157- 

RADIO  MEASUREMENTS  AND  MISCELLANEOUS. 

61.  The  Principles  of  Electric  Wave  Telegraphy  and  Telephony;  J.  A.  Fleming; 

3d  ed.,  1916. 

62.  Les  Oscillations  Electriques;  C.  Tissot;  1910. 

63.  Radiotelegraphisches  Praktikum;  H.  Rein;  1912. 


326  Circular  of  the  Bureau  of  Standards 

64.  Wireless  Telegraphy;  J.  Zenneck  (translated  into  English  by  A.  E.  Seelig); 


65.  Wireless  Telegraphy  and  Telephony,  A  Handbook;  W.  H.  Eccles;  1916. 

66.  Radio  Communication;  J.  Mills;  1917. 

67.  Standardization  Rules,  Institute  of  Radio  Engineers;  1915. 

WAVE  LENGTH. 

71.  Die   Frequenzmesser  und    Dampfungsmesser  der  drahtlosen   Telegraphic;   E. 

Nesper;  1907. 

72.  Standard  Wave  Length  Circuits;  A.  Campbell;  Phil.  Mag.,  18,  p.  794;  1909. 

Electrician,  64,  p.  612;  1910. 

73.  Calibration  of  Wavemeters;  G.  W.  O.  Howe;  Electrician,  69,  p.  490;  1912. 

74.  Wavemeter  Standardization;  Diesselhorst;  Elektrotechnische  Zs.,  29,  p.   703; 

1908. 

75.  Pointer-Type  Wavemeter;  Feme  and  Carpentier;  Jahrb.  d.  drahtl.  Tel.,  5,  p. 

106;  1911. 

76.  Practical  Uses  of  the  Wavemeter  in  Wireless  Telegraphy;  J.  O.  Mauborgne;  1914. 

77.  Oval  Diagram  for  Wave  Length  Calculations;  W.  H.  Eccles;  Electrician,  76, 

p.  388;  1915. 

CAPACITY. 

81.  Square-  Plate  Condenser  for  Uniform  Scale  of  Wave  Lengths;  C.  Tissot;  Journal 

de  Physique,  2,  p.  719;  1912. 

82.  Rotary  Condenser  for  Uniform  Scale  of  Wave  Length;  W.  Duddell;  Jour.  I.  E.  E., 

52,  p.  275;  1914. 

83.  A.-c.     Resistance  of  Condensers;  Fleming  and  Dyke;  Electrician,  68,  pp.  1017, 

1060,  1912;  69,  p.  10,  1912.     G.  E.  Bairsto;  Electrician,  76,  p.  53,  1915. 

84.  Calculation  of  Capacity   Using  Method  of   Images;   "Alternating   Currents";. 

A.  Russell;  Vol.  I,  chaps.  5  and  6;  1914. 

INDUCTANCE. 

91.  The  Effects  of  Distributed  Capacity  of  Coils  Used  in  Radiotelegraphic  Circuits; 

F.  A.  Kolster;  Proc.  I.  R.  E.,  1,  p.  19;  1913. 

92.  Distributed  Capacity  of  Single-Layer  Solenoids;  J.  C.  Hubbard;  Phys,  Review, 

9,  p.  529;  1917. 

93.  Development  of  Inductance  Formulas;   "Alternating  Currents";  A.   Russell; 

Vol.  I,  chaps.  2   and  3;  1914.     "Absolute  Measurements  in  Electricity  and 
Magnetism";  A.  Gray;  Vol.  II,  part  i,  chap.  6. 

CURRENT  MEASUREMENT. 

101.  Thermoelements  for  High-Frequency  Measurements;  Dowse;  Electrician,  65, 

p.  765;    1910. 

102.  Hot-Strip  Ammeters  for  Large  High-Frequency  Currents;  R.  Hartmann-Kempf; 

Elektrotechnische  Zs.,  32,  p.  1134;  1911.     G.  Eichhorn;  Jahrbuch  d.  drahtl. 
Tel.,  5,  p.  517;  1912. 

103.  High-Frequency  Current  Transformer;  Campbell  and  Dye;  Proc.  Royal  Soc., 

90,  p.  621;  1914. 

104.  Use  of  Iron  in  High-Frequency  Current  Transformer;  McLachlan;  Electrician, 

78,  p.  382;  1916. 

105.  Use  of  Galvanometer  in  Audion  Plate  Circuit;  L.  E.  Whittemore;  Phys.  Review, 

9,  p.  434;  1917. 

106.  Measurement  of  Signal  Intensity  with  Crystal  Detector;  J.  L.  Hogan;  (Marconi) 

Year-Book  of  Wireless  Telegraphy,  p.  662;  1916. 


Radio  Instruments  and  Measurements  327 

107.  Measurements  With  Crystal  and  Telephone;  J.  Zenneck;  Proc.  I.  R.  E.,  4,  p. 

363;  1916. 

108.  Current  Measurement  With  the  Audion;  L.  W.  Austin;  Jour.  Wash.  Acad.  Sci- 

ences, 6,  p.  81;  1916.    Proc.  I.  R.  E.,  4,  p.  251;  1916.     Electrician,  78,  p. 
465;  1917.    Proc.  I.  R.  E.,  6,  p.  239;  1917. 

HIGH-FREQUENCY  RESISTANCE. 

in.  Skin  Effect  in  Round  Wires;  Lord  Rayleigh;  Phil.  Mag.,  pp.  382,  469;  1886;  Sci. 
Papers,  Vol.  II,  pp.  486,  495.  Skin  Effect  in  Round  Wires;  Lord  Kelvin; 
Math,  and  Phys.  Papers,  Vol.  Ill,  p.  491;  1889. 

112.  Skin  Effect  in  Stranded  Conductors  to  Oscillatory  Currents;  F.  Dolezalek;  Ann. 

der  Phys.,  (4),  12,  p.  1142;  1903. 

113.  Passage  of  High-Frequency  Current  Through  Coils;  M.  Wien;  Ann.  der  Phys., 

(4),  14,  p.  i;  1904. 

114.  Long  Solenoids  at  High  Frequencies,  Mathematical  Theory;  A.  Sommerfeld; 

Ann.  der  Phys.,  (4),  15,  p.  673,  1904;  (4),  24,  p.  609,  1907. 

115.  Calorimetric  Measurements  of   High-Frequency   Resistance  of   Solenoids;   T. 

Black;  Ann.  der  Phys.,  19,  p.  157;  1906. 

116.  Measurements  on  Stranded  Conductors;  R.  Lindemann;  Verh.  deutsch.  Phys. 

Gesel.,  11,  p.  682;  1909. 

117.  Theory  for  Stranded-Conductor  Solenoids;  M6ller;  Ann.  der  Phys.,  36,  p.  738, 

1911;  and  Jahr.  draht.  Tel.,  9,  p.  32,  1914. 

118.  Measurements  on  Single  and  Multiple  Layer  Coils;  Esau;  Ann.  der  Phys.,  84, 

p.  57;  1911. 

119.  Skin  Effect  in  Flat  Coils  and  Short  Cylindrical  Coils;  Lindemann  and  Hiiter; 

Verh.  deutsch.  Phys.  Ges.,  15,  p.  219;  1913. 

120.  The  Alternating-Current  Resistance  of  Long  Coils  of  Stranded  Wire,  Theory; 

Rogowski;  Arch.  f.  Elect.,  3,  p.  264;  1915. 

121.  Bibliography,  and  Measurements  on  Wires  and  Strips;  Kennelly,  Laws,  and 

Pierce;  Proc.  A.  I.  E.  E.,  34,  p.  1749;  1915. 

122.  Bibliography,  and  Measurements  on  Solid  and  Stranded  Conductors;  Kennelly 

and  Affel;  Proc.  I.  R.  E-,  4,  p.  523;  1916. 

123.  High-Frequency  Resistance  of  Multiply-Stranded  Insulated  Wire;     G.  W.  O. 

Howe;  Proc.  Royal  Society  London,  93,  p.  468;  1917. 

124.  The  Accuracy  of  High-Frequency  Resistance  Measurements;  S.  Loewe,  Jahr- 

buch  d.  Drahtlosen  Telegraphic,  7,  p.  365;  1913. 

ELECTRON  TUBES. 

131.  Theory  of  Thermionic  Emission;  O.  W.  Richardson;  Phil.  Trans.,  202,  p.  516; 

1903. 

132.  Audion  Detector  and  Amplifier;  L.  De  Forest;  Electrician,  73,  p.  842;  1914. 

Elec.  World,  65,  p.  465;  1914. 

133.  Theory  of  Electron  Tubes;  I.  Langmuir;  Phys.  Review,  2,  p.  450;  1913.     Proc. 

I.  R.  E.,  3,  p.  261;  1915. 

134.  Operating  Features  of  the  Audion,  Amplification,  etc.;  E.  H.  Armstrong;  Elec. 

World,  64,  p.  1149;  1914.     Proc.  I.  R.  E.,  8,  p.  215,  1915;  5,  p.  145,  1917. 

135.  Characteristic  Curves,  and  Use  as  Source  of  High  Frequency  Current;  J.  Beth- 

enod;  La  Lumiere  Electrique,  35,  pp.  25,  225;  1916. 

136.  Generalized  Equations  for  Audions;  M.  Latour;  La  Lumiere  Electrique,  Dec. 

30,  1916.     Electrician,  78,  p.  280;  1916. 

137.  Characteristics  of  Audion  Tubes  Used  in  Radiotelegraphy ;  G.  Vallauri;  L'Elet- 

trotecnica,  4,  Nos.  3,  4,  18,  and  19;  1917. 

138.  Use  of  Pliotron  to  Produce  Extreme  Frequencies,  Currents,  and  Voltages; 

W.  C.  White;  General  Electric  Review,  19,  p.  771,  1916;  20,  p.  635,  1917. 


328  Circular  of  the  Bureau  of  Standards 

MISCELLANEOUS  SOURCES  OF  HIGH-FREQUENCY  CURRENT. 

141.  Disturbing  Short  Waves  in  Buzzer  Circuits;  S.  Loewe;  Jahrb.  d.  drahtl.  Tel., 

6,  p.  325;  1912. 

142.  Production  of  Undamped  Oscillations;  M.  Wien;  Jahrb.  d.  drahtl.  Tel.,   1, 

p.  474;  1908.     Physikaltsche  Zs.,  11,  p.  76;  1910. 

143.  Impulse  Excitation  Transmitter;  E.  W.  Stone;  Proc.  I.  R.  E.,  4,  p.  233,  1916; 

&•  P-  133.  I9I7- 

144.  Frequency  Multipliers;  A.  N.  Goldsmith;  Proc.  I.  R.  E.,  3,  p.  55;i9i5-     W.  H. 

Eccles;  Electrician,  72,  p.  944;  1914. 

145.  High-Frequency  Alternator  of  Induction  Type;  General  Electric  Review,  16, 

p.  16;  1913. 

146.  High-Frequency  Alternator  Employing  Rotating  Magnetic  Fields;  R.  Gold- 
schmidt;  Electrician,  66,  p.  744;  1911.     T.  R.  Lyle;  Electrician,  71,  p.   1004; 


147.  Duddell  Arc;  W.  Duddell;  Jour.  Rontgen  Soc.,  4,  p.  i;  1907. 

148.  Arc  generator  for  laboratory  purposes;  F.  Kock,  Phys.  Zeitschr.,  12,  p.  124;  1911. 

149.  Impact  excitation  of  undamped  waves;  E.  L.  Chaffee;  Jahrb.  d.  drahtl.  Tel.  7, 

p.  483;  1913.     Proc.  Amer.  Ac.  Arts  &  Sci.,  47,  No.  9;  p.  267;  1911. 


UNITS  AND  INSTRUMENTS 

151.  Units  of  Weight  and  Measure;  Circular  No.  47;  1914. 

152.  Electric  Units  and  Standards;  Circular  No.  60;  1916.     International  System  of 
Electric  and  Magnetic  Units;  J.  H.  Bellinger;  Bull.,  13,  p.  599;  1916  (S.  P.  292). 

153.  Electrical  Measuring  Instruments;  Circular  No.  20;  2d  ed.,  1915. 

154.  Fees  for  Electric,  Magnetic,  and  Photometric  Testing;  Circular  No.  6;  7th  ed., 

1916. 

ELECTRICAL  PROPERTIES  OF  MATERIALS 

161.  Copper  Wire  Tables;  Circular  No.  31;  3d  ed.,  1914. 

162.  Electric  Wire  and  Cable  Terminology;  Circular  No.  37;  2d  ed.,  1915. 

163.  Insulating  Properties  of  Solid  Dielectrics;  H.  L.  Curtis;  Bull.,  11,  p.  359;  1914 

(S.  P.  234). 

CAPACITY  AND  INDUCTANCE 

171.  The  Testing  and  Properties  of  Electric  Condensers;  Circular  No.  36;  1912. 

172.  Formulas  and  Tables  for  the  Calculation  of  Mutual  and  Self  Inductance;  Rosa 

and  Grover;  Bull.,  8,  p.  i;  1911  (S.  P.  169). 

173.  Various  papers  on  inductance  calculations;  see  Circular  No.  24,  "Publications 

of  the  Bureau  of  Standards." 

174.  The  Absolute  Measurement  of  Capacity;  Rosa  and  Grover;  Bull.,  1,  p.  153;  1904 

(S.  P.  10). 

175.  Measurement  of  Inductance  by  Anderson's  Method,  Using  Alternating  Currents 

and  a  Vibration  Galvanometer;  Rosa  and  Grover;  Bull.,  1,  p.  291;  1905  (S.  P. 

14). 

176.  The  Simultaneous  Measurement  of  the  Capacity  and  Power  Factor  of  Con- 

densers; F.  W.  Grover,  Bull.,  3,  p.  371;  1907  (S.  P.  64). 

177.  Mica  Condenser  as  Standards  of  Capacity;  H.  L.  Curtis,  Bull.,  6,  p.  431;  1910 

(S.  P.  137). 

178.  The  Capacity  and  Phase  Difference  of  Paraffined  Paper  Condensers  as  Func- 

tions of  Temperature  and  Frequency;  F.  W.  Grover;  Bull.,  7,  p.  495;  1911 
(S.  P.  166). 


Radio  Instruments  and  Measurements  329 

179.  The  Measurement  of  the  Inductances  of  Resistance  Coils;  Grover  and  Curtis; 

Bull.,  8,  p.  455!  19"  (S.  P.  175)- 

180.  Resistance  Coils  for  Alternating  Current  Work;  Curtis  and  Grover;  Bull.,  8,  p. 

495;  1911  (S.  P.  177). 

181.  A  Variable  Self  and  Mutual  Inductor;    Brooks  and  Weaver;  Bull.,  13,  p.  569; 

1916  (S.  P.  290). 

RADIO  SUBJECTS 

191.  The  Influence  of  Frequency  Upon  the  Self-Inductance  of  Coils;  J.  G.  Coffin; 

Bull.,  2,  p.  275;  1906  (S.  P.  37). 

192.  The  Influence  of  Frequency  on  the  Resistance  and  Inductance  of  Solenoidal 

Coils;  L.  Cohen;  Bull.,  4,  p.  161;  1907  (S.  P.  76). 

193.  The  Theory  of  Coupled  Circuits;  L.  Cohen;  Bull.,  5,  p.  511;  1909  (S.  P.  112). 

194.  Coupled  Circuits  in  which  the   Secondary  has  Distributed  Inductance  and 

Capacity;  L.  Cohen;  Bull.,  6,  p.  247;  1909  (S.  P.  126). 

195.  High-Frequency  Ammeters;  J.  H.  Bellinger;  Bull.,  10,  p.  91;  1913  (S.  P.  206). 

196.  Direct-Reading  Instrument  for  Measuring  Logarithmic  Decrement  and  Wave 

Length  of  Electromagnetic  Waves;  F.  A.  Kolster;  Bull.,  11,  p.  421;  1914 

(S.  P.  235). 

197.  Effect  of  Imperfect  Dielectrics  in  Field  of  Radiotelegraphic  Antennas;  J.  M. 

Miller;  Bull.,  13,  p.  129;  1916  (S.  P.  269). 

PUBLICATIONS  OF  THE  UNITED  STATES  NAVAL  RADIOTELEGRAPHIC  LABARATORY 
IN  THE  BULLETIN  OF  THE  BUREAU  OF  STANDARDS. 

201.  Detector  for  Small  Alternating  Currents  and  Electrical  Waves;  L.  W.  Austin; 

Bull.,  1,  p.  435:  1905  (S.  P.  22). 

202.  The  Production  of  High- Frequency  Oscillations  from  the  Electric  Arc;  L.  W. 

Austin;  Bull.,  3,  p.  325;  1907  (S.  P.  60). 

203.  Some  Contact  Rectifiers  of  Electric  Currents;  L.  W.  Austin;  Bull.,  5,  p.  133; 

1908  (S.  P.  94). 

204.  A  Method  of  Producing  Feebly  Damped  High-Frequency  Electrical  Oscillations 

for  Laboratory  Measurements;  L.  W.  Austin;  Bull.,  5,  p.  149;  1908  (S.  P.  95). 

205.  The  Comparative  Sensitiveness  of  Some  Common  Detectors  of  Electrical  Oscilla- 

tions; L.  W.  Austin;  Bull.,  6,  p.  527;  1910  (S.  P.  140). 

206.  The  Measurement  of  Electric  Oscillations  in  the  Receiving  Antenna;  L.  W. 

Austin;  Bull.,  7,  p.  295;  1911  (S.  P.  157). 

207.  Some  Experiments  with  Coupled  High-Frequency  Circuits;  L.  W.  Austin;  Bull., 

7,  p.  301;  1911  (S.  P.  158). 

208.  On  the  Advantages  of   a  High  Spark   Frequency  in  Radiotelegraphy;  L.  W. 

Austin;  Bull.,  5,  p.  153;  1908  (S.  P.  96). 

209.  Some  Quantitative  Experiments  in  Long    Distance    Radiotelegraphy;  L.  W. 

Austin;  Bull.  7,  p.  315;  1911  (S.  P.  159). 

210.  Antenna  Resistance ;  L.  W.  Austin;  Bull.,  9,  p.  65;  1912  (S.  P.  189). 

211.  The  Energy  Losses  in  Some  Condensers  Used  in  High-Frequency  Circuits; 

L.  W.  Austin;  Bull.,  9,  p.  73  (S.  P.  190). 

212.  Quantitative  Experiments  in  Radiotelegraphic  Transmisssion,  L.  W.  Austin; 

Bull.,  11,  p.  69;  1914  (S.  P.  226). 

213.  Note  on  Resistance  of  Radiotelegraphic  Antennas;  L.  W.  Austin;  Bull.  12,  p. 

465;  1915  (S.  P.  257). 


330 


Circular  of  the  Bureau  of  Standards 


APPENDIX  3.— SYMBOLS  USED  IN  THIS  CIRCULAR 


l?=magnetic  induction. 
c=  velocity  of  light=2.9982Xio10  cm 

per  second. 

C=electrostatic  capacity. 
d=diameter. 

e=instantaneous  electromotive  force. 
JE=effective  electromotive  force. 
E0=maximum  electromotive  force. 

£3==electric  field  intensity. 

/=frequency. 

F=force. 

,j/^=  magnetomotive  force. 
7/=magnetic  field  intensity. 

t  =  instantaneous  current. 

7=effective  current. 
70=maximum  current. 


p — 


£=coupling  coefficient. 
/C=dielectric  constant. 
2=length. 

L=self-inductance. 

w=mass. 

M=  mutual  inductance. 

p=  instantaneous  power. 
P=average  power. 

^=quantity  of  electricity. 

Special  symbols  are  denned  where  used  in  Part  III  and  elsewhere. 


r =distance  from  a  point. 
/?=resistance. 
R=reluctance. 
j=length  along  a  path. 
S=area. 
/=time. 

T=period  of  a  complete  oscillation. 
7>=velocity. 

F=potential  difference  of  a  condenser. 
w=instantaneous  energy. 
W= average  energy. 
AT=reactance. 
Z=impedance. 
5=logarithmic  decrement. 
t=base    of    napierian    logarithms  == 

2.71828. 
0=phase  angle. 
X=wave  length. 
^t=permeability. 

volume  resistivity. 

magnetic  flux. 

phase  difference. 

27rXfreqiiency. 

microfarad. 

micromicrofarad. 

microhenry. 


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Bidg.  400,  Richmond  Field  Static 


be  recharged  by  bringin 

recharges  may  be  made  4 
day!  prior  to  due  date. 

DUE  AS  STAMPED  BELOW 


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